Distance d'un point à une ligne

Supposons qu'il y ait une ligne et un point dans un plan cartésien. Que peut-on faire pour relier la ligne et le point et définir une caractéristique qui les mettrait en relation ?

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    L'un de ces moyens fondamentaux de les relier est la distance qui les sépare, et c'est ce que nous allons tenter de découvrir dans cet article : la distance d'un point à une droite.

    Définition de la distance entre un point et une ligne

    Analysons ce problème à l'aide du diagramme ci-dessous.

    Distance d'un point à une ligne, une ligne et un point A avec quelques distances sur celui-ci, StudySmarter

    Une ligne l et un point A avec quelques-unes des distances possibles entre eux, StudySmarter Originals

    Si l'on te demandait de trouver la distance entre le point A et la ligne l, ce serait vraiment ambigu, car il y a plusieurs façons, en fait infinies, de relier le point et la ligne.

    Il faut donc être précis quant à la distance demandée. Certaines de ces distances sont représentées par les segments verts dans le diagramme ci-dessus.

    Mais la seule distance spécifique qui peut être nommée et quantifiée est la distance la plus courte entre le point et la ligne.

    La distance la plus courte entre la ligne et le pointest représentée par le segment de ligne rose dans le diagramme ci-dessus .

    La distance entre un point et une ligne est donnée par la distance la plus courte entre eux.

    Le segment de droite le plus court est perpendiculaire à la droite elle-même. Étant donné que tout autre segment de droite formera un angle aigu ou obtus avec la droite, la distance la plus courte ne sera possible que lorsqu'elle sera perpendiculaire à la droite.

    Cette distance peut également être considérée comme le chemin le plus court par lequel le point A peut être amené à la ligne l.

    Mais comment déterminer la distance entre un point et une ligne en utilisant l'équation d'une ligne et les coordonnées du point ? Voyons comment nous pouvons trouver une telle formule.

    Formule de calcul de la distance entre un point et une ligne

    Soit D une ligne droite dont l'équation est donnée par ax+by+c=0a,b ne sont pas simultanément 0, et un point A x0,y0 à l'extérieur de la droite, c'est-à-dire n'appartenant pas à la droite.

    Le but est de trouver la distance la plus courte entre la droite D et le point P. Soit le point où le segment de droite le plus court coupe la droite, Q dont les coordonnées sont x1,y1.

    La distance entre le point et la ligne D est la même que la longueur du segment de ligne formé par les points A et Q ou la distance entre eux. Nous pouvons utiliser la formule de la distance pour y parvenir, mais nous avons besoin de connaître les coordonnées de Q en termes de x0 and y0 pour cela.

    Distance entre un point et une ligne, La distance entre un point et une ligne, StudySmarterLa distance entre un point et une ligne, StudySmarter Originals

    Rappelle que le gradient d'une ligne dont l'équation est ax+by+c=0 est donné par m1=-ab. Maintenant, le segment de droite AQ est perpendiculaire à la droite et sa pente sera donc m2=ba. En effet, le produit des pentes de deux droites perpendiculaires est toujours égal à -1, soitm1m2=-1.

    Nous avons maintenant la pente de la ligne joignant AQ et les coordonnées d'un point A sur cette ligne. À l'aide de ces informations, nous pouvons maintenant former l'équation de la ligne AQ,

    y-y0=m2x-x0

    y-y0=bax-x0

    ay-ay0=bx-bx0

    Puisque Q se trouve sur cette ligne, nous pouvons substituer x1,y1 par x,y pour trouver les inconnues x1 and y1.

    ay1-ay0=bx1-bx0

    Mais Q se trouve également sur la ligne ax+by+c=0ax+by+c=0{"x":[[197,198,203],[213,214,215,219,219,219,216,215,213,212,205,202,200,190,175,169,167,165,163,160,158,157,155,155,155,156,159,162,167,169,179,182,193,196,208,212,215,222,225,230,237,238,244,246,249,251,254,257,260,261,267,269,275,277],[316,319,323,327,330,336,343,345,349,352,354,354,352,351,345,342,337,327,324,315,314,311,310],[395,395,396,396,396,396,392,390,388,383,380,377,372,369,367,365,362,360,358,358,358,360,361,366,369,378,382,394,402,406],[467,467,468,471,473,481,485,488,495,499,509,515,522,524,527,532,536,538,539],[492,492,492,493,493,494,494,494,494,494,493,493,493],[587,586,585,585,584,584,582,581,579,578,575,574,573,573,573,572,572,572,573,575,577,579,583,585,593,597,605,608,612,618,621,625,629,631,632,634,635,635,633,631,623,616,609,600,596,583,579,572,570],[685,686,686,687,687,687,686,685,685,685,684,684,687,689,690,694,696,698,701,706,708,710,716,719,721,726,729,733,737,740,741,742,743,743,743,742,742,742,741,741,741,741,741,740,739,738,736,735,733,730,726,724,717,714,706,703,696,693],[793,794,795,796,799,801,807,814,818,829,836,839,845,847,850,851,852,853],[827,827,827,826,827,829,831,832,833,834,834,834,833,833,833,833,834,835,836],[942,942,942,942,941,940,937,932,929,919,916,906,900,897,894,891,891,891,892,894,896,901,903,910,917,925,934,938],[982,984,989,997,1003,1009,1012,1021,1023,1027,1031,1032,1033],[975,975,976,978,980,982,986,989,993,999,1002,1008,1015,1020,1023,1029,1030],[1075,1076,1076,1076,1075,1074,1071,1069,1067,1066,1063,1062,1062,1062,1062,1062,1065,1068,1069,1073,1075,1077,1083,1085,1091,1096,1101,1104,1111,1113,1117,1118,1120,1120,1120,1120,1119,1118,1115,1112,1110,1108,1102,1098,1094,1088,1086,1081,1076,1073,1072]],"y":[[369,369,368],[345,344,342,326,324,319,311,309,307,306,304,304,304,309,330,342,347,354,360,373,380,385,398,402,406,407,408,406,404,402,393,390,378,373,361,357,354,349,347,346,346,348,358,362,373,379,391,403,412,416,424,426,429,428],[321,319,317,316,316,317,322,325,332,339,352,357,369,373,386,390,397,409,410,416,417,417,417],[330,329,327,323,322,320,320,321,323,328,331,335,343,348,353,358,368,374,387,394,401,408,411,418,419,420,420,417,413,409],[356,355,355,354,354,352,351,351,350,350,350,349,348,348,347,344,344,343,343],[327,327,329,334,341,345,360,364,376,379,389,391,394],[261,261,263,265,280,287,305,317,329,335,351,360,364,371,373,376,380,381,383,383,383,380,377,375,368,365,360,359,358,357,357,358,360,361,363,366,367,369,376,379,386,391,395,399,400,402,402,400,398],[327,327,326,328,332,339,347,355,363,366,376,378,382,383,384,384,384,383,382,379,377,374,368,365,362,355,352,345,339,334,333,331,331,334,336,345,350,353,363,373,386,398,405,418,424,430,441,446,456,464,470,473,478,479,478,477,470,467],[364,364,364,364,364,364,364,364,364,364,364,364,364,364,364,363,363,363],[337,336,335,333,331,332,334,337,344,353,368,373,383,393,398,403,405,410,411],[314,313,312,311,309,308,307,309,310,320,324,341,355,367,379,391,400,403,409,412,413,415,416,417,416,414,410,408],[339,339,339,338,337,336,335,334,334,333,333,333,334],[379,380,381,381,382,382,381,381,380,379,378,377,375,374,374,373,372],[307,305,304,303,303,306,314,323,335,343,365,372,377,389,393,397,404,411,413,417,419,420,422,422,421,417,412,408,395,390,374,368,350,344,338,327,322,317,308,304,301,299,296,295,295,297,299,303,307,311,315]],"t":[[0,20,36],[204,220,235,281,282,290,308,315,316,323,336,340,352,360,374,387,390,391,403,407,407,419,425,436,440,441,453,461,473,475,487,492,503,507,520,523,524,536,540,553,560,572,578,587,590,591,602,607,618,624,635,640,652,661],[1017,1026,1036,1043,1052,1057,1073,1074,1086,1090,1102,1107,1119,1124,1135,1140,1152,1157,1172,1174,1186,1190,1191],[1346,1351,1359,1373,1376,1387,1402,1407,1407,1419,1424,1424,1432,1440,1441,1452,1457,1467,1475,1486,1491,1502,1507,1519,1524,1535,1541,1552,1557,1569],[1754,1769,1776,1786,1791,1802,1807,1810,1819,1824,1836,1840,1852,1857,1857,1870,1878,1882,1887],[2043,2078,2082,2092,2102,2108,2119,2124,2136,2141,2152,2157,2158],[2392,2404,2419,2424,2436,2441,2449,2457,2469,2476,2489,2491,2504,2507,2508,2519,2524,2525,2536,2541,2553,2558,2569,2576,2589,2591,2604,2607,2608,2616,2624,2636,2641,2641,2653,2657,2658,2670,2677,2688,2691,2704,2708,2719,2724,2736,2741,2753,2757],[3088,3096,3104,3124,3135,3141,3153,3158,3169,3175,3190,3191,3204,3208,3208,3220,3224,3225,3237,3241,3241,3254,3258,3258,3270,3274,3275,3289,3291,3304,3308,3320,3325,3337,3341,3353,3358,3358,3366,3374,3389,3391,3404,3408,3408,3420,3425,3425,3437,3442,3453,3458,3470,3475,3483,3491,3504,3510],[3822,3831,3843,3853,3858,3858,3870,3875,3887,3894,3903,3911,3916,3925,3933,3933,3941,3953],[4064,4068,4078,4087,4102,4121,4125,4137,4141,4153,4158,4170,4175,4188,4195,4200,4208,4220,4225],[4451,4460,4462,4471,4476,4487,4494,4507,4508,4522,4525,4537,4541,4554,4558,4571,4575,4587,4592,4592,4604,4608,4609,4621,4625,4637,4642,4650],[4923,4930,4934,4935,4947,4950,4958,4970,4975,4987,4992,4992,5004],[5175,5184,5188,5189,5195,5203,5208,5209,5221,5225,5226,5238,5242,5254,5259,5271,52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satisfera donc à l'équation de la ligne D, d'où l'équation suivante

    ax1+by1+c=0ax1+by1+c=0{"x":[[203,204,205,205,206,207,207,208,208,208,208,205,204,202,201,197,195,192,187,184,181,174,171,166,161,156,154,149,148,146,146,148,150,153,155,157,160,165,168,171,177,180,183,190,192,198,203,208,212,216,218,222,222,225,225,226,226,227,227,227,227,227,228,229],[276,277,283,289,297,303,310,318,320,327,329,332,332,332,330,327,323,317,312,306,301,299,296],[351,349,346,343,340,337,335,329,326,323,321,314,312,308,308,307,308,310,314,319,322,331,335,339,347,351],[398,398,398,398,398,398,398,398,397,397,397,396,396,396,396],[439,440,443,446,447,449,457,460,469,472,475,478,483,490,494],[468,467,466,466,466,466,466,465,465,462,460,459,459,458,458,458,458,459],[550,550,550,550,550,550,549,548,547,546,543,543,541,540,540,539,539,538,538,539,540,542,545,547,551,553,555,560,566,571,579,584,588,590,592,592,589,587,580,578,569,567,564,559,556,554,552,549,548],[629,629,630,630,630,631,630,630,630,630,630,631,632,635,636,641,642,644,648,651,653,658,661,663,665,669,671,673,677,679,681,683,684,684,684,683,682,680,680,678,676,675,674,673,672,669,669,667,666,664,662,660,658,656,653,651,646,644],[711,711,711,712,712,712,713,714,715,715,715,716,716,717,717,717,717,717,719,720],[749,748,746,749,750,758,760,764,771,774,778,785,789,793,796,801,804,807,809,811],[771,773,774,775,776,776,777,777,775,774,771,770,769,767,766,765,765,764,764,764,764,765,766,768],[916,915,904,901,895,886,882,877,873,863,859,856,849,845,840,838,838,841,845,848,859,863,878,884,899,904],[965,968,970,974,982,985,991,994,997,1003,1006,1008,1011,1013,1014],[942,942,942,943,946,948,951,962,966,980,983,994,999,1002,1007,1010,1013,1016,1020],[1069,1069,1069,1068,1068,1065,1064,1063,1061,1059,1058,1057,1057,1057,1059,1063,1067,1068,1075,1078,1081,1087,1090,1099,1102,1104,1109,1114,1115,1119,1120,1122,1124,1124,1124,1123,1120,1115,1110,1105,1099,1092,1087,1081,1076,1070]],"y":[[263,262,261,260,260,257,255,249,247,240,237,231,228,227,226,224,224,224,226,227,230,236,240,248,258,270,277,295,302,313,316,322,324,325,325,325,325,322,320,317,312,309,305,298,294,287,281,275,271,269,268,270,272,281,285,289,297,302,306,309,317,320,324,333],[255,253,248,246,245,244,245,247,249,254,257,266,272,276,284,291,299,307,314,320,325,327,329],[253,254,254,256,260,262,266,273,278,282,287,301,305,318,322,328,331,336,340,343,344,346,346,346,345,344],[336,337,338,340,344,347,350,353,359,362,365,371,374,376,380],[284,285,286,287,287,288,289,289,289,289,289,288,287,286,285],[256,256,255,258,260,271,275,290,295,308,315,321,325,330,335,337,339,340],[203,207,211,221,233,240,247,258,263,268,279,283,290,294,297,302,304,309,311,312,311,310,308,307,304,303,302,300,298,296,296,296,296,298,301,302,308,310,316,318,324,326,327,330,331,331,331,331,330],[255,256,258,259,261,270,278,282,286,292,295,304,306,312,313,315,315,315,313,312,310,306,303,300,297,289,285,281,273,269,263,258,255,253,252,255,257,267,271,283,293,305,311,317,323,340,344,353,357,365,370,375,376,377,378,378,375,374],[335,336,337,339,340,341,346,349,352,355,360,363,366,374,379,384,387,390,391,390],[288,288,289,290,290,290,290,290,289,289,289,288,287,287,286,284,284,283,282,281],[252,252,252,252,252,253,259,262,274,279,292,297,301,309,312,316,319,325,328,330,334,336,338,339],[241,240,239,239,242,246,249,253,257,269,274,280,292,299,311,321,329,335,341,343,347,348,348,347,343,341],[272,272,272,272,272,272,271,271,270,270,269,269,269,269,269],[318,319,320,321,321,321,321,318,317,314,313,311,311,310,310,309,309,309,308],[232,233,237,243,247,264,270,281,293,306,311,317,327,331,338,344,349,351,354,355,355,355,353,347,344,341,334,325,320,310,306,300,289,284,274,264,256,247,239,232,227,224,222,222,223,230]],"t":[[0,7,11,19,27,39,45,56,62,72,77,90,94,94,106,111,111,123,127,128,139,144,144,156,162,173,177,189,195,206,211,223,227,240,244,244,256,261,261,273,277,278,290,294,294,306,312,322,328,340,344,356,361,373,377,378,390,394,398,406,411,411,422,430],[732,749,752,754,761,771,777,789,795,806,811,819,827,839,844,856,861,872,877,889,894,895,906],[1092,1108,1113,1114,1120,1128,1128,1139,1144,1145,1156,1161,1173,1178,1189,1194,1195,1206,1212,1223,1228,1239,1244,1247,1256,1261],[1554,1560,1570,1578,1589,1594,1595,1606,1611,1611,1623,1628,1628,1640,1644],[1964,1978,1981,1986,1989,1995,2006,2012,2023,2028,2028,2040,2045,2056,2061],[2240,2245,2249,2270,2278,2290,2296,2306,2312,2323,2328,2340,2345,2356,2361,2362,2373,2376],[2609,2637,2645,2657,2662,2662,2674,2678,2678,2690,2695,2695,2707,2712,2713,2724,2729,2740,2745,2757,2762,2774,2778,2790,2795,2795,2807,2812,2820,2828,2840,2846,2857,2862,2873,2879,2890,2895,2907,2912,2924,2928,2929,2937,2945,2948,2957,2962,2966],[3248,3261,3264,3265,3271,3278,3290,3295,3297,3307,3312,3324,3328,3340,3345,3357,3362,3362,3370,3379,3379,3390,3395,3396,3407,3412,3412,3424,3431,3431,3444,3445,3454,3463,3473,3490,3496,3507,3512,3524,3529,3540,3545,3546,3558,3564,3574,3579,3579,3591,3595,3607,3612,3613,3624,3629,3641,3646],[3943,3948,3953,3954,3962,3966,3974,3979,3979,3991,3995,3996,4008,4012,4024,4029,4041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    Les deux droites ci-dessus se coupent en Q et peuvent donc être résolues simultanément afin de déterminer les inconnues x1 and y1En écrivant la première équation en termes de x1,

    x1=-c-by1a

    En substituant l'expression de x1 dans ay1-ay0=bx1-bx0on obtient

    ay1-ay0=b-c-by1a-bx0-bc-b2y1-a2y1+a2y0-abx0=0

    En résolvant poury1 nous obtenons

    ay1=ay0-bac+by1-bx0{"x":[[58,58,57,57,56,55,52,49,47,42,39,36,34,32,31,29,29,30,32,34,38,46,53,57,59,59,58,58,58,58,59,60,62,64,65,68,69,73,77],[89,89,89,89,89,90,91,91,93,94,96,103,108,112,118,122,123,124,124,123,121,120,120,119,119,118,118,118,117,115,114,113,111,108,106,105,102,100,95,93,92,88],[140,140,139,139,138,138,137,136,136,137],[177,176,175,176,177,182,189,191],[178,180,181,184,190,191,195,197,199],[301,301,301,301,300,296,291,289,286,282,280,279,278,278,279,283,287,289,294,296,300,301,302,302,302,302,302,303,305,308],[326,326,326,325,325,325,324,324,324,324,324,324,326,327,329,335,342,346,347,350,352,352,352,352,350,348,347,346,346,346,346,346,345,344,344,342,341,339,336,333,330,327,325],[367,367,367,367,368,369,369,369,368,368,368,368,368,370,371,375,380,381,382,383,382,379,377,372,370,369],[428,426,426,425,427,429,431,438,444,451,454],[515,515,516,516,516,516,516,516,515,514,513,513,512,512,510,510,511,513,516,518,519,522,525,526,528,529,529,525,520,516,512,511],[504,500,499,501,502,508,511,517,524,527,531,539,541,543],[517,519,519,519,519,517,517,514,507,499,495,494,494,495,498,501,503,509,514,519,520,521,522,522,522,523,526,528,529,531],[594,593,593,593,592,591,589,588,583,579,577,575,575,576,577,580,582,588],[627,627,626,625,623,620,616,614,614,616,618,622,630],[651,651,654,656,658,660,665],[662,661,660,659,659,659,659,659,659,659,659,659,659,659],[694,694,694,694,694,693,692,692,691,690,689,688,688,687,686,685,685,687,689,691,694,698,699,701,703,703,703,703,701,699,696,694,693,691],[727,727,727,728,728,728,728,728,728,727,726,726,729,734,738,743,750,751,753,753,753,752,751,750,748,747,745,744,742,741,740,740,739,739,738,736,732,731,729,726,725,722],[768,767,766,766,766,766,767,767,767,768,768,768,769],[802,802,802,803,804,805,808,809,813,813,814,812,810,809,804,802,796,794,791],[862,861,860,863,872,874,879,885,886,888,887,886,885,884],[928,928,927,927,926,926,926,925,925,924,923,922,921,919,917,916,916,918,923,926,928,933,934,938,939,939,939,937,933,928,925,924,923,922,920,919],[970,970,969,969,970,974,978,981,983,985,986,986,983,982,978,978,977],[1000,998,998,997,996,993,991,990,987,986,983,983,985,990,998,1005,1009],[1027,1027,1026,1025,1024,1024,1026,1027,1032,1035,1037,1038,1040,1038,1036,1034,1030,1028,1027]],"y":[[171,170,169,167,166,165,163,163,163,166,170,175,178,183,186,194,198,199,200,200,198,191,183,177,175,176,180,184,186,192,194,197,200,200,200,198,196,193,189],[172,169,168,167,168,171,173,175,186,190,193,192,189,185,178,172,170,165,164,164,166,170,172,178,181,187,190,195,209,218,227,230,236,243,244,245,245,245,242,241,239,234],[206,205,205,208,215,217,222,227,231,232],[183,183,182,182,181,181,181,181],[196,196,196,195,193,192,191,191,190],[181,178,177,176,173,172,173,174,178,184,186,190,192,193,196,196,194,193,190,189,185,185,184,185,186,187,188,190,192,192],[179,178,177,177,176,175,174,175,178,179,183,184,190,191,194,195,191,187,185,180,176,174,173,171,170,170,175,184,189,193,201,205,210,218,221,228,232,234,235,234,232,230,226],[226,225,224,223,222,222,223,226,230,231,234,236,238,240,240,238,232,230,227,223,222,220,220,219,220,220],[190,190,189,189,189,189,189,189,189,188,188],[163,158,157,156,155,154,157,158,163,170,173,175,178,180,188,192,193,191,189,187,186,185,185,185,186,188,191,194,196,197,197,196],[226,227,227,227,227,227,227,227,226,226,226,225,225,225],[268,267,266,265,263,261,260,260,261,268,273,278,279,281,281,281,280,277,274,271,271,271,274,275,278,281,282,282,281,280],[173,172,171,170,169,168,168,170,177,187,190,203,217,219,222,227,228,232],[184,182,181,180,180,184,190,196,199,200,201,201,199],[191,190,190,189,189,189,188],[184,183,183,182,183,184,185,186,188,189,192,197,198,199],[169,166,165,163,162,163,168,169,173,175,181,183,185,188,190,193,195,195,195,194,193,192,192,192,192,193,195,196,198,201,202,202,202,201],[183,182,181,179,178,177,178,180,182,187,194,199,202,202,201,199,195,194,192,191,190,189,189,189,190,190,192,193,196,197,201,209,213,224,231,236,241,242,242,239,238,233],[220,218,218,217,218,221,223,224,226,229,230,232,233],[163,162,161,161,163,164,170,172,182,186,195,207,210,214,220,223,229,231,234],[190,190,190,190,190,190,190,190,190,190,190,190,190,190],[166,162,161,159,159,158,159,159,164,165,172,175,183,191,197,198,200,199,195,193,192,191,191,191,191,194,196,200,203,206,207,208,208,208,206,206],[185,184,183,182,181,182,183,185,186,189,194,196,201,202,205,206,207],[183,181,180,181,183,187,189,190,194,196,202,204,208,210,210,210,209],[222,223,226,227,231,235,236,236,235,234,232,231,227,223,221,221,220,220,220]],"t":[[0,8,8,18,24,35,51,58,68,74,83,91,101,108,108,124,135,142,151,158,168,185,201,218,224,241,251,258,268,274,285,291,301,308,318,324,325,335,347],[858,859,868,884,901,908,923,925,943,950,959,975,985,991,1008,1018,1025,1035,1052,1068,1085,1091,1102,1108,1118,1125,1125,1135,1141,1152,1158,1171,1175,1185,1192,1194,1202,1209,1218,1225,1225,1235],[3034,3068,3075,3092,3109,3119,3126,3136,3153,3169],[3666,3676,3693,3717,3726,3736,3753,3761],[3905,3943,3943,3951,3959,3970,3976,3976,3987],[4466,4475,4479,4484,4503,4520,4536,4543,4553,4559,4570,4576,4576,4589,4593,4609,4620,4626,4639,4643,4654,4661,4670,4689,4693,4693,4704,4709,4726,4743],[4985,4993,5010,5010,5020,5026,5043,5060,5070,5076,5087,5093,5104,5110,5120,5137,5153,5160,5170,5176,5187,5193,5204,5210,5220,5237,5253,5269,5276,5287,5293,5293,5304,5310,5310,5324,5327,5337,5343,5352,5360,5370,5376],[5837,5841,5845,5852,5868,5894,5905,5910,5927,5927,5935,5946,5960,5971,5976,5987,6004,6010,6021,6027,6037,6043,6054,6060,6071,6075],[13649,13655,13658,13671,13689,13696,13707,13713,13730,13740,13757],[14263,14271,14276,14277,14281,14297,14313,14323,14330,14340,14346,14347,14358,14363,14374,14390,14407,14423,14430,14441,14446,14457,14463,14474,14480,14497,14507,14524,14538,14555,14563,14574],[14949,14952,14957,14977,14980,14990,14997,15007,15013,15014,15025,15030,15041,15047],[15343,15352,15356,15363,15380,15390,15397,15407,15424,15441,15457,15463,15472,15480,15491,15497,15509,15514,15530,15541,15547,15558,15574,15580,15591,15610,15624,15630,15631,15640],[16183,16189,16197,16208,16225,16230,16241,16247,16264,16275,16280,16291,16308,16314,16314,16325,16331,16342],[17445,17448,17464,17475,17492,17509,17524,17539,17557,17564,17578,17581,17598],[17792,17806,17839,17848,17860,17864,17881],[17997,17998,18006,18014,18049,18057,18064,18067,18076,18082,18092,18108,18115,18125],[18277,18280,18286,18290,18298,18331,18348,18351,18358,18367,18375,18381,18382,18393,18398,18409,18425,18442,18449,18463,18465,18481,18492,18498,18511,18516,18526,18531,18543,18559,18575,18581,18593,18598],[19017,19024,19032,19051,19065,19075,19101,19107,19115,19126,19142,19159,19176,19182,19193,19198,19215,19226,19232,19243,19248,19260,19265,19276,19282,19282,19294,19299,19310,19316,19326,19332,19343,19348,19360,19365,19382,19392,19398,19410,19415,19426],[19741,19756,19766,19773,19791,19798,19810,19815,19827,19832,19843,19848,19856],[20075,20081,20093,20107,20115,20126,20132,20143,20149,20162,20165,20177,20182,20182,20194,20199,20210,20215,20223],[20987,20988,21007,21032,21049,21060,21066,21077,21082,21094,21099,21110,21116,21121],[21312,21317,21325,21332,21344,21349,21360,21366,21377,21382,21394,21400,21411,21427,21432,21444,21449,21482,21499,21510,21516,21527,21532,21544,21550,21561,21566,21577,21593,21599,21610,21616,21616,21628,21632,21644],[22010,22015,22032,22049,22066,22083,22093,22099,22111,22116,22127,22133,22144,22150,22161,22166,22166],[22348,22355,22367,22391,22399,22410,22416,22416,22426,22433,22445,22449,22461,22477,22483,22494,22500],[22952,23025,23041,23050,23061,23077,23083,23095,23100,23111,23116,23128,23133,23150,23161,23167,23178,23183,23188]],"version":"2.0.0"}

    En développant les parenthèses et en réarrangeant les termes, nous obtenons

    ay1=a2y0-bc-b2y1-abx0a

    En multipliant les deux côtés par aon obtient

    a2y1+b2y1=a2y0-abx0-bc{"x":[[86,87,87,87,86,84,83,81,80,74,71,65,63,60,59,59,60,62,67,69,75,78,83,85,90,90,92,92,93,93,94,95,96,98,98,102,103,108,110],[115,111,110,112,113,117,120,121,123,123,124,125,126,126,124,122,118,114,114,117,125,130],[158,158,158,159,159,160,160,160,160,159,159,159,159,159,160,165,172,174,178,182,183,185,186,185,184,183,182,181,180,180,179,179,179,178,177,176,175,173,170,169,166,165,163,160],[204,204,204,204,205,205,205,205,205,204,203,203],[246,245,244,243,242,242,243,248,249,252,255,259,260,261],[256,255,255,255,255,254,252,251],[319,319,319,318,317,314,313,312,310,310,309,309,309,310,312,313,317,324,325,331,333,334,334,333,328,326,323,318,316,312],[348,349,349,349,349,349,351,352,352,355,356,356,354,351,349,348,346,350,352,360,363],[387,387,387,387,388,389,390,390,390,391,392,393,395,399,401,406,409,411,412,414,415,415,415,414,413,411,411,409,408,408,407,407,404,403,401,399,397,396,392,391],[441,440,440,440,440,441,441,441,442,442],[485,484,483,485,491,497,503,505,507],[487,486,485,486,487,491,495,499,505,511,512,515],[604,604,604,604,604,602,598,596,595,593,590,589,587,586,588,590,593,597,600,606,610,613,614,614,614,614,615],[627,627,627,627,627,627,627,629,632,635,636,636,634,633,631,630,627,626,625,627,637,642],[672,672,672,671,671,671,671,672,672,673,673,674,675,677,678,680,685,688,691,692,694,695,695,694,693,692,691,690,687,684,681,680,679,677,675,673,673],[704,704,704,705,707,708,708,709,709,709,709,709,709,709,710,711,712,714,716,717,717,717,717,709,708],[749,748,751,752,757,759,764,769,770,772,771],[811,812,813,813,814,814,814,815,815,812,810,807,803,800,799,797,794,794,795,798,799,802,804,807,808,811,813,813,814,815,815,816,817,822,823],[847,847,846,846,846,846,847,848,850,850,850,850,849,847,843,842,841,843,844,848,849,854,855,858,860,862,862,862,862,859,858,855,854,851,848,847,846],[892,891,890,890,890,889,892,896,899,899,900,900,900,899,898,894,893,895,901,903,908,909,913,914,915,914,912,909,909,908,908,908,910,913],[931,930,930,930,929,929,929,929,930,931,933,934,935,936,937,937,937,937,934,933,930,929],[970,968,967,968,974,975,978,983,985,987,988,990],[1025,1025,1025,1025,1025,1025,1025,1025,1025,1025,1024,1023,1021,1021,1018,1017,1017,1016,1017,1018,1021,1023,1027,1030,1032,1035,1036,1036,1036,1036,1035,1033,1032,1030,1027,1025,1023,1022,1021,1021],[1074,1075,1076,1076,1077,1078,1078,1078,1077,1074,1072,1069,1067,1065,1063,1064,1067,1071,1073,1080]],"y":[[219,218,216,214,210,208,208,208,208,212,215,223,227,235,239,242,242,243,242,241,236,234,228,226,220,219,217,218,218,221,222,228,235,238,239,241,241,238,236],[161,157,155,153,153,153,153,153,154,155,156,157,162,163,169,171,177,181,182,182,180,178],[202,200,199,197,195,195,197,199,202,209,211,213,217,218,222,223,219,218,215,211,209,205,200,199,198,198,199,201,205,214,225,228,235,245,249,251,255,257,260,260,260,259,258,255],[243,240,239,240,244,245,250,251,254,258,261,262],[219,219,219,219,219,220,221,223,223,223,223,223,223,222],[217,215,214,217,220,223,231,237],[199,196,195,194,197,209,215,219,229,231,237,239,242,243,243,242,241,238,238,237,238,240,241,244,247,248,249,250,250,250],[184,180,179,176,175,174,174,174,175,178,184,185,190,196,199,201,205,205,205,202,202],[220,218,217,216,215,216,219,221,224,230,235,236,237,237,236,233,229,226,224,221,216,215,214,213,213,219,222,228,235,238,241,251,263,265,270,270,269,268,265,263],[260,258,257,258,261,265,267,271,273,276],[224,224,224,224,223,223,223,223,223],[241,241,241,241,241,240,240,239,238,237,237,237],[237,236,234,232,230,229,231,232,234,236,242,244,249,252,252,251,248,245,242,238,236,237,241,242,244,246,247],[205,203,201,200,197,195,194,193,194,196,201,205,209,210,213,214,217,218,220,220,218,216],[225,223,222,221,220,221,222,226,232,233,235,237,238,239,239,239,236,234,231,230,226,224,223,225,231,235,241,243,252,261,269,271,273,274,272,271,270],[264,262,261,260,259,259,261,261,265,266,269,271,273,274,274,274,273,271,268,266,263,262,261,259,259],[232,232,232,232,231,231,231,231,231,231,232],[232,230,229,228,227,226,225,222,221,220,220,222,225,228,230,231,235,239,239,239,239,236,235,233,232,230,230,231,231,234,235,237,239,241,241],[204,203,202,200,199,198,198,199,203,205,211,216,218,227,236,237,240,239,239,237,236,234,233,232,232,233,234,235,237,241,242,244,244,245,245,244,242],[220,220,219,218,217,217,218,220,223,224,226,228,229,231,233,236,237,235,230,229,225,223,220,219,218,218,221,225,227,231,232,235,238,240],[250,250,251,252,253,254,257,258,260,260,260,260,260,258,257,255,254,253,251,251,250,249],[232,232,232,232,232,232,232,232,232,232,232,232],[210,207,206,205,203,202,203,204,209,212,220,223,231,234,241,244,245,246,246,246,243,242,240,239,239,239,240,241,242,244,245,247,248,249,250,250,250,250,248,247],[229,228,228,227,226,224,223,222,222,225,227,230,233,235,240,243,245,245,245,245]],"t":[[0,4,13,31,47,64,70,70,81,86,95,103,113,120,130,137,147,153,163,170,180,186,199,204,214,220,229,245,247,253,265,270,286,299,303,314,320,332,337],[604,613,629,646,654,664,675,6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a2+b2y1=a2y0-abx0-bc{"x":[[72,73,73,74,75,76,76,75,74,71,66,58,53,52,51,53,54,55,59,67,76,81,81,81,80,80,80,80,81,84,86],[108,105,104,103,103,104,105,109,113,115,115,115,114,110,106,105,104,107,109,115,118],[133,131,134,140,144,148,153,155],[148,147,146,145,145,145,144,143,142,142,143],[182,182,182,182,182,182,182,182,182,182,182,182,181,181,181,181,181,181,181,184,186,189,193,195,196,197,196,194,193,192,190,189,188,186],[212,211,210,210,209,209,210,211,212,213,216,219,221,222,222,219,218,215,215,213,215,217,219,221,227],[34,34,34,34,34,34,34,33,32,31,29,27,24,23,21,24,29,32],[251,251,251,252,255,256,258,260,261,264,264,264,259,255,250,245],[293,293,293,293,293,294,295,296,296,296,296,296,296,299,302,307,309,316,317,320,323,324,324,324,323,321,319,318,317,317,317,317,317,316,315,313,310,306,305],[361,361,361,360,360,360,358,357,357,357,357],[405,404,405,408,413,414,417],[402,400,399,401,407,409,413,416,424,427],[535,536,536,535,533,532,531,525,520,519,516,516,518,525,526,528,531,534,537,538,539,539,540,540,541,541,542,543,546,549,550],[562,561,560,559,559,559,559,561,565,569,570,570,568,565,563,561,560,560,564,572,576],[597,597,596,596,596,596,597,597,597,597,597,598,604,607,611,617,618,620,620,619,618,617,616,615,614,613,613,612,612,611,611,609,607,606,602,601,598,597],[638,638,638,638,639,639,638,638,638,639,640,642,646,647,648,649,650,649,648,646,641,640],[674,673,673,674,677,681,687,690,696],[748,748,748,748,747,745,743,740,739,734,731,730,729,729,730,732,736,739,740,743,745,746,749,750,751,753,754,754,755],[784,783,782,782,781,780,780,779,779,779,778,777,777,776,775,775,775,776,778,779,783,784,787,791,793,793,792,788,783,782,780,778,778],[820,820,819,819,818,819,822,824,825,826,826,827,826,825,823,822,821,822,823,824,826,830,832,838,843,844,843,843,841,840,839,838,838,838,841,842,843],[861,862,862,863,864,864,865,865,864,864,863,862,862,863,863,864,866,867,871,873,873,873,871,867,866,864],[897,896,899,901,903,908,910,917,918,922,923,924],[954,954,954,955,955,956,958,957,956,954,951,951,950,951,952,955,958,961,964,966,966,966,966,963,962,960,957,956,953,952,951],[998,998,998,998,998,998,996,995,994,990,989,986,985,983,983,985,986,990,996,999,1006,1008,1010]],"y":[[204,204,203,203,202,199,195,193,193,193,196,204,213,215,217,219,219,219,218,212,206,200,201,203,206,207,210,214,215,216,216],[165,164,163,162,161,160,160,159,159,162,163,164,169,174,178,180,180,180,179,177,177],[210,210,210,209,209,209,208,207],[201,200,199,198,199,200,206,212,219,220,225],[186,183,182,181,180,179,181,185,189,194,196,200,202,205,208,210,212,214,215,215,214,213,212,212,213,215,218,220,221,221,222,222,222,221],[163,161,161,160,159,157,157,155,155,154,153,153,154,158,160,165,167,172,173,177,178,178,177,177,176],[186,185,184,183,182,181,180,180,180,180,182,186,194,198,209,222,228,230],[173,170,169,169,171,172,174,179,182,191,194,200,210,217,223,228],[183,181,180,179,178,178,180,186,195,197,202,206,207,209,208,206,204,199,197,194,189,187,185,183,181,181,185,190,197,200,208,211,217,224,226,229,229,226,225],[225,223,222,223,224,228,235,238,240,243,244],[193,193,193,193,193,193,193],[210,211,212,212,212,211,210,210,208,208],[200,198,197,195,193,193,193,194,199,201,205,210,212,210,209,208,206,203,201,201,201,202,205,206,209,210,211,212,213,213,212],[168,166,166,165,164,163,162,160,159,160,163,167,172,175,179,182,183,184,184,181,180],[191,190,190,189,188,189,190,193,194,199,202,206,207,206,203,198,197,195,194,194,194,196,197,201,203,209,219,227,231,233,235,238,238,237,235,234,230,226],[225,224,223,222,223,225,229,230,233,235,235,234,231,230,229,227,226,223,222,221,221,221],[201,201,200,200,200,199,199,199,199],[209,206,205,204,202,201,201,201,202,206,210,214,215,216,217,217,216,214,213,212,212,211,211,211,211,212,214,215,216],[186,181,180,179,177,177,176,177,183,185,190,197,199,207,213,214,216,216,215,214,212,211,209,208,207,209,212,216,218,218,218,217,216],[203,202,202,201,200,201,203,204,206,206,208,211,212,214,217,217,218,216,214,213,212,207,205,201,197,197,199,200,203,205,209,210,213,216,218,218,218],[226,226,225,225,224,223,222,223,226,227,229,232,233,235,236,236,236,235,233,230,227,226,224,222,222,222],[203,203,204,204,204,204,204,203,203,203,203,203],[188,186,185,183,182,181,187,194,199,206,213,215,216,216,214,212,209,207,207,207,208,209,211,214,215,216,217,218,218,218,217],[204,201,200,199,198,196,196,196,196,198,199,203,205,210,211,215,216,217,218,218,217,217,216]],"t":[[0,6,21,29,39,54,71,89,96,108,112,129,146,157,163,172,179,179,189,206,223,239,263,273,279,289,297,305,313,323,339],[693,702,713,730,740,756,763,773,790,807,813,813,823,840,846,856,863,879,894,896,907],[1202,1216,1238,1256,1263,1273,1290,1294],[1424,1436,1439,1446,1480,1480,1496,1517,1521,1531,1540],[1684,1688,1697,1699,1707,1723,1738,1747,1757,1763,1774,1780,1780,1790,1796,1797,1807,1813,1824,1840,1846,1857,1874,1880,1889,1896,1913,1923,1930,1941,1946,1957,1963,1971],[2174,2179,2182,2189,2197,2213,2224,2230,2230,2241,2247,2257,2274,2293,2297,2307,2314,2326,2331,2341,2360,2363,2364,2374,2380],[2909,2915,2924,2930,2941,2947,2958,2964,2974,2980,2991,2997,3007,3014,3024,3039,3047,3055],[3453,3462,3465,3472,3489,3497,3497,3508,3518,3522,3532,3541,3547,3558,3564,3574],[4024,4028,4032,4042,4058,4064,4075,4091,4112,4115,4125,4131,4141,4147,4158,4164,4175,4181,4191,4197,4208,4214,4225,4231,4242,4258,4275,4291,4297,4308,4314,4325,4331,4342,4347,4356,4372,4389,4397],[4798,4801,4823,4856,4865,4873,4889,4898,4908,4914,4915],[5399,5407,5441,5457,5464,5474,5479],[5565,5573,5581,5607,5625,5627,5631,5642,5648,5659],[6376,6382,6386,6398,6415,6425,6431,6442,6457,6465,6475,6492,6509,6535,6540,6540,6549,6565,6576,6582,6592,6598,6609,6616,6626,6632,6632,6643,6659,6665,6673],[6826,6830,6833,6840,6848,6859,6865,6876,6893,6909,6926,6943,6948,6959,6965,6982,6982,6993,6998,7015,7021],[7403,7416,7419,7423,7432,7449,7459,7465,7476,7482,7499,7509,7536,7540,7550,7565,7576,7582,7593,7599,7615,7626,7632,7643,7649,7660,7676,7693,7699,7699,7710,7715,7732,7743,7749,7760,7765,7780],[7924,7931,7950,7974,8007,8024,8032,8043,8049,8066,8076,8082,8093,8099,8099,8111,8116,8124,8133,8143,8149,8160],[8447,8451,8466,8476,8482,8495,8510,8518,8523],[8887,8898,8904,8908,8924,8933,8943,8949,8961,8966,8983,8993,8999,9011,9016,9024,9041,9049,9060,9066,9066,9078,9083,9083,9094,9099,9116,9127,9133],[9265,9276,9278,9283,9293,9299,9310,9327,9343,9349,9362,9366,9377,9383,9399,9410,9416,9433,9444,9449,9461,9467,9477,9494,9510,9518,9535,9549,9566,9577,9583,9594,9599],[9912,9916,9933,9944,9966,9983,10000,10011,10016,10028,10033,10044,10049,10061,10066,10078,10094,10111,10116,10117,10128,10133,10146,10150,10166,10177,10196,10200,10211,10217,10228,10234,10241,10258,10266,10280,10281],[10510,10517,10525,10533,10551,10558,10575,10600,10616,10627,10633,10644,10650,10662,10666,10667,10679,10684,10692,10708,10717,10727,10733,10750,10761,10766],[13130,13159,13210,13217,13228,13234,13243,13251,13259,13268,13278,13284],[13461,13467,13472,13478,13496,13509,13537,13551,13562,13568,13584,13595,13601,13618,13629,13645,13662,13679,13684,13701,13701,13712,13718,13734,13745,13751,13762,13768,13779,13784,13796],[13991,13995,14000,14002,14009,14018,14034,14045,14051,14063,14068,14080,14085,14096,14101,14113,14118,14130,14134,14146,14151,14163,14166]],"version":"2.0.0"}

    Nous allons maintenant diviser par a2+b2{"x":[[213,214,214,214,214,214,214,214,212,210,208,206,200,198,191,189,183,179,176,175,174,174,174,175,176,179,181,182,204,207,215,218,227,231,232,234,238,241,243,247,249,251,256,258,260,266,269,275,280],[288,286,283,282,281,281,280,279,279,280,281,285,289,293,295,297,300,302,307,309,309,310,310,309,308,305,302,299,296,294,293,292,292,295,298,303,306,316,319,323,331],[374,375,376,378,379,387,391,402,409,414,421,424,427,436,442],[408,409,412,413,414,417,419,421,421,421,421,421,420,420,420],[503,503,503,503,504,505,505,505,505,505,505,504,504,503,503,503,504,504,505,506,508,510,511,513,523,527,531,535,539,542,543,544,543,542,537,534,529,527,523,517,516,514],[553,553,553,553,554,556,559,561,563,570,575,579,583,584,588,589,590,590,588,585,581,579,577,575,572,571,570,570,571,575,577,584,593,598,609,615]],"y":[[303,299,297,294,291,289,286,284,281,279,277,277,278,279,287,290,303,313,324,329,334,342,344,346,347,348,348,348,332,328,318,314,306,306,308,310,319,326,331,337,340,343,346,347,348,348,348,346,342],[220,218,216,215,214,213,210,207,205,202,200,197,195,193,193,193,193,194,198,202,205,212,215,221,225,232,238,245,250,255,257,262,263,265,265,265,265,263,263,262,261],[317,317,317,317,317,316,315,313,312,311,310,310,309,308,307],[294,293,294,295,297,302,309,317,326,330,337,341,344,349,354],[216,217,221,224,239,251,262,272,283,292,296,310,313,324,327,333,334,335,336,336,334,333,332,331,327,327,327,328,329,330,332,333,337,338,343,346,349,350,351,351,350,347],[200,198,195,194,191,190,188,186,186,184,184,184,186,187,190,191,195,196,201,206,210,212,214,217,220,222,223,225,226,226,226,225,223,222,220,218]],"t":[[0,8,13,15,21,30,35,35,46,52,63,68,80,85,96,102,113,118,130,135,136,146,152,152,163,169,169,180,213,218,230,235,257,260,261,271,280,285,296,302,303,313,318,319,330,335,336,344,352],[618,623,628,635,635,646,652,662,669,679,685,696,702,713,719,719,729,735,749,752,761,769,779,785,786,796,802,813,819,829,835,846,852,863,869,879,885,896,902,902,916],[1150,1161,1164,1165,1170,1179,1186,1196,1203,1213,1219,1219,1230,1235,1257],[1431,1442,1461,1469,1469,1480,1486,1494,1502,1513,1519,1519,1530,1536,1546],[1782,1788,1803,1812,1819,1830,1836,1845,1852,1863,1869,1883,1886,1896,1902,1913,1919,1919,1930,1936,1949,1952,1953,1961,1983,1986,1996,2003,2013,2019,2030,2037,2047,2052,2063,2069,2080,2086,2097,2102,2113,2119],[2376,2383,2390,2403,2403,2413,2419,2420,2430,2436,2447,2453,2463,2469,2480,2486,2497,2503,2513,2519,2530,2536,2536,2547,2553,2553,2564,2569,2580,2586,2586,2597,2603,2614,2619,2624]],"version":"2.0.0"}pour obtenir

    y1=a2y0-abx0-bca2+b2

    En remplaçant ce résultat par x1=-c-by1a pour déterminer x1on obtient

    x1=-ca-baa2y0-abx0-bca2+b2{"x":[[39,38,37,36,35,33,34,39,43,45,51,56,54,49,43,36,34,35],[83,82,81,81,79,75,74,70,67,64,62,60,58,61,70,76],[107,107,107,107,107,107,107,107,107,106,105,105],[146,145,145,144,146,147,151,158,160,162,164],[151,150,152,153,159,166,169,173],[238,237,236,235,236,238,240,244,246,252,257,259,260],[297,297,297,296,296,295,293,292,285,280,276,278,280,282,286,288,295],[270,268,267,266,265,266,269,274,289,293,300,309,311,313,311,309],[299,300,300,300,300,300,299,297,294,292,289,288,284,282,282,282,284,285,289,294,296,300,301,303,305,306,306,306,307,309],[347,346,345,344,343,344,349,358,369,372],[418,418,418,418,417,417,416,416,415,414,413,413,413,415,419,420,423,424,427,429,430,431,431,430,429,426,425,422,419,418,415,415],[406,405,406,407,411,416,427,439,446,446,445],[428,430,430,430,431,431,430,429,427,424,423,419,416,415,414,413,413,413,415,420,421,425,426,428,430,431,432,433,435,438,442],[510,510,511,511,510,506,500,494,490,489,489,490,495,505,509],[570,570,570,569,569,566,563,562,555,549,546,547,551,554,558,559,561,565,568,570,571,571,572,572,572,574,575],[594,593,593,591,590,590,591,595,600,602,602,602,599,593,592,590,592,597,599,605],[623,623,623,623,623,622,622,622,625,627,631,637,642,644,645,643,642,641,640,640,639,639,639,637,637,635,634,631,625,623],[660,660,660,661,661,660,660,659,659,659,660,661,663,670,673,673,671,670,668,661,660,658,657],[690,689,688,689,692,693,696,698,700,704,709,710],[737,738,738,739,740,740,740,740,739,735,731,726,725,723,722,722,723,725,726,727,731,735,736,739,741,742,743,744,745,747,748],[774,774,773,773,772,772,773,773,773,772,772,770,768,768,769,772,777,778,781,783,784,785,785,781,777,773,772,771,771],[807,806,805,805,804,804,806,809,810,812,814,815,815,815,813,811,810,809,812,814,815,819,820,825,827,828,828,828,825,824,823,822,822,825],[841,842,843,843,844,845,845,845,845,845,844,843,844,846,847,850,850,852,852,851,848,846],[882,881,880,881,882,887,894,895,898,900],[925,925,925,925,925,925,925,925,924,923,922,922,922,923,925,929,933,934,935,936,937,936,936,933,931,927,926,925,923],[959,957,957,956,955,954,953,952,951,951,950,948,948,948,948,949,951,957,959,963],[612,614,622,627,645,660,675,715,726,748,783,795,830,842,870,886,899,904,911,913,915,914,912,909,905,902],[690,691,692,692,693,693,692,689,687,682,677,672,671,670,668,668,668,669,672,673,677,679,680,683,685,689,690,693,694,695,695,696,698,699],[720,718,717,717,719,720,724,726,726,727,727,725,724,721,720,717,718,724,730],[752,751,750,749,753,760,762,766,770,771],[763,763,763,763,762,762,762,762,761,761,761,761,761,761],[805,804,804,803,802,802,802,802,802,801,801,801,799,799,798,799,803,804,806,809,810,812,812,813,812,811,808,805,803,801,799],[829,829,828,828,828,829,830,832,835,837,837,838,837,835,832,831,829,827,829,836,839,844],[1001,1000,1000,1000,1002,1004,1007,1009,1010,1013,1013,1008,1006,998,995,986,980,977,971,969,965]],"y":[[128,128,128,127,127,125,124,123,123,123,124,130,137,144,149,153,152,151],[125,125,125,124,124,123,124,126,129,133,135,139,145,150,149,148],[152,151,150,149,150,152,156,158,165,172,179,181],[125,125,124,124,123,123,123,123,123,123,123],[139,139,138,138,137,135,135,134],[132,132,132,132,132,132,132,132,132,131,130,130,130],[124,122,120,119,118,118,117,117,122,128,135,140,140,140,140,140,139],[168,169,169,169,169,169,170,170,169,169,168,167,167,166,166,167],[201,200,199,198,196,195,193,192,192,193,195,196,202,208,209,212,213,213,212,209,208,206,205,204,204,206,207,208,209,209],[149,149,149,148,148,148,148,149,148,147],[122,117,116,115,114,117,126,129,134,142,144,151,154,154,152,152,150,150,150,150,150,151,152,154,155,156,157,157,158,158,157,156],[179,178,178,178,178,177,177,177,177,178,178],[210,209,208,207,206,202,201,199,198,198,198,201,205,207,208,211,213,214,214,214,213,212,212,211,211,212,214,217,218,217,215],[122,119,117,113,110,109,116,133,150,168,174,191,207,214,214],[129,127,126,125,122,121,122,123,129,137,143,143,141,139,136,134,133,130,129,130,131,132,134,135,138,141,141],[101,99,98,96,95,94,94,94,97,99,103,105,109,115,116,117,117,117,117,116],[124,122,121,120,123,127,131,136,139,140,140,138,134,131,128,127,127,128,133,141,154,157,164,171,173,174,174,173,168,166],[152,150,149,150,154,155,158,162,163,165,166,166,166,161,157,153,151,150,149,148,148,148,148],[132,131,130,130,130,130,130,130,130,130,129,129],[135,133,132,131,128,127,125,124,123,122,124,127,129,134,135,137,139,139,139,139,137,133,132,131,130,130,130,133,135,137,137],[107,105,104,102,101,103,109,111,115,121,123,128,133,134,134,132,130,129,128,128,128,130,132,136,138,139,139,138,137],[123,123,123,122,122,121,121,122,122,123,126,127,129,130,132,135,135,137,135,134,132,130,129,125,123,122,121,122,124,126,128,131,134,136],[142,142,142,141,141,140,141,142,143,145,148,152,153,153,152,149,148,145,142,140,139,138],[122,122,122,122,122,122,122,122,122,122],[108,106,105,104,103,106,112,113,119,126,129,130,132,132,131,129,128,128,128,129,129,131,132,134,135,136,136,137,136],[127,125,124,123,122,121,121,1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    En réduisant à un dénominateur commun, nous obtenons

    x1=-c(a2+b2)-ba2y0+ab2x0+b2caa2+b2

    En simplifiant, on obtient

    x1=-a2c-b2c-a2by0+ab2x0+b2ca(a2+b2)

    En simplifiant davantage en éliminant les termes similaires, nous obtenons

    x1=b2x0-aby0-aca2+b2

    Nous avons maintenant obtenu les coordonnées du point Q en fonction des constantes que nous connaissons,

    Qb2x0-aby0-aca2+b2,a2y0-abx0-bca2+b2

    Nous pouvons maintenant calculer la distance entre A et Q en utilisant la distance, qui n'est rien d'autre que la distance entre le point et la ligne , comme nous l'avons vu précédemment. Notons-la par d et appliquons la formule de la distance,

    d2=x1-x02+y1-y02

    En substituant pour x1,y1 nous obtenons

    d2=b2x0-aby0-ac-a2x0-b2x0a2+b2+a2y0-aby0-bc-a2y0-b2y0a2+b2

    En simplifiant davantage, nous obtenons

    d2=a2ax0+by0+c2+b2ax0+by0+c2(a2+b2)2

    d2=a2+b2ax0+by0+c2a2+b22

    d2=ax0+by0+c2a2+b2

    En prenant la racine carrée des deux côtés, on obtient ,

    d=±ax0+by0+ca2+b2

    Puisque d est la distance, il ne peut pas être négatif, nous rejetons donc la racine négative, ce qui nous donne,

    d=ax0+by0+ca2+b2

    Mais il y a encore une possibilité lorsque le numérateur ax0+by0+c est négatif. Pour éviter qu'il soit négatif, il faut prendre son module,

    d=ax0+by0+ca2+b2

    Nous ne rencontrons pas ce problème puisque le dénominateur est une somme de carrés de nombres non nuls, il sera donc toujours positif.

    Pour écrire la même expression sous une forme plus pratique (et plus facile à retenir), définissons l'équation de la droite comme suit fx,y=ax+by+c pour obtenir fx0,y0=ax0+by0+cce qui nous donne

    d=|fx0,y0|a2+b2

    Appliquons maintenant cette formule à travers quelques exemples.

    Calculer la distance d'un point à une ligne

    Le calcul de la distance d'une ligne à un point est un processus relativement simple. L'équation d'une ligne doit être donnée ainsi que le point jusqu'auquel la distance doit être calculée.

    Trouve la distance entre la ligne 4x+3y-2=0 et le point 2,4.

    Solution

    En comparant l'équation donnée à la forme générale, nous obtenons a=4, b=3 et c=-2, et x0=2, y0=4.

    En rappelant la formule de la distance et en substituant toutes les valeurs correspondantes, nous obtenons ,

    d=ax0+by0+ca2+b2=42+34-242+32=185

    Ainsi, la distance entre la ligne 4x+3y-2=0 et 2,4 est de 185 unités.

    Trouve la distance entre la ligne 5x-2y=0{"x":[[208,204,202,199,195,190,187,178,175,166,164,156,154,152,149,147,146,144,143,142,141,140,139,139,138,137,135,134,132,131,129,128,126,125,123,123,123,123,123,123,124,125,127,132,135,140,142,151,154,164,168,178,181,185,190,192,194,198,199,202,202,202,200,199,197,193,190,185,179,173,170,163],[260,260,261,263,266,271,274,280,284,287,295,298,304,307,315,317,317,315,313,304,301,297,291,287,283,278,275,273],[346,348,348,349,349,348,345,341,339,332,329,321,319,316,315,315,317,318,321,323,325,330,337,345,353,357],[414,414,415,418,422,427,430,442,445,452,458,466,468,469,470],[508,508,508,508,511,514,519,522,525,528,534,539,543,545,548,549,550,550,548,545,543,535,532,524,519,515,513,512,510,510,510,511,513,514,519,522,528,536,543,552,560,563,573,577],[618,619,620,621,622,622,623,623,623,624,624,625,625,626,628,629,630,634,637,640,644,646,649,654,657,660,664,666,671,675,679,680,683,684,684,683,682,680,678,676,673,671,669,668,667,666,666,665,665,664,663,661,660,656,654,644,640,638,633,632],[757,758,760,762,766,771,774,785,788,794,796,799,804],[732,733,738,741,748,760,764,777,786,794,798,802,805,812],[889,890,890,890,888,887,884,883,880,879,877,875,875,873,873,873,874,877,879,886,889,897,903,907,914,917,920,927,931,933,939,942,944,949,951,954,957,958,958,956,955,949,942,936,932,927,923,915,907,900,897,895,891,889,888,886]],"y":[[227,229,229,230,231,233,234,238,239,241,242,243,243,243,243,242,242,242,242,242,242,243,245,247,249,254,263,271,278,282,293,297,307,310,318,320,322,325,326,327,327,327,325,320,316,312,310,304,303,298,298,298,298,299,302,304,307,312,315,325,332,336,344,348,352,360,363,370,376,380,382,384],[296,294,290,286,283,280,279,277,276,276,276,276,278,280,289,298,301,312,316,329,332,336,343,346,349,353,355,356],[278,277,275,275,274,274,275,278,279,285,288,297,301,312,318,324,330,332,336,338,340,342,343,343,342,341],[293,294,294,294,294,294,293,292,292,291,290,289,289,289,289],[267,266,264,263,260,258,257,256,256,256,257,259,262,264,268,273,276,279,286,293,296,307,311,320,326,331,333,335,338,340,341,342,342,342,342,342,341,339,337,334,332,331,328,327],[269,269,269,269,270,272,277,284,288,303,308,321,324,328,333,336,338,342,344,345,345,345,345,342,340,338,333,331,325,319,313,310,303,302,298,298,300,306,309,316,324,334,345,352,358,371,377,391,401,412,422,429,432,440,442,445,442,441,436,433],[302,302,302,303,303,303,303,303,303,303,303,303,304],[351,351,352,351,350,346,345,342,341,340,340,340,340,339],[256,256,254,253,256,257,266,270,280,287,293,309,315,329,343,356,361,376,381,390,393,398,399,399,397,395,393,388,384,380,370,365,359,347,342,329,317,306,300,284,279,267,260,256,255,254,254,255,258,262,265,267,272,274,277,281]],"t":[[0,10,12,14,25,35,39,51,55,68,72,85,89,89,101,105,118,122,123,130,139,151,156,156,168,172,185,189,201,207,218,223,235,239,251,256,256,264,272,284,289,301,306,318,322,334,340,351,356,368,373,385,389,390,402,406,406,417,422,431,439,452,456,456,468,472,473,485,489,501,507,514],[846,852,857,858,864,872,884,889,889,901,906,906,918,923,940,948,956,967,973,984,989,989,1001,1006,1006,1014,1023,1023],[1188,1195,1199,1201,1206,1218,1223,1235,1240,1251,1256,1268,1273,1284,1289,1301,1306,1318,1323,1323,1335,1339,1351,1356,1368,1370],[1625,1639,1656,1668,1674,1684,1690,1701,1707,1718,1723,1735,1739,1740,1748],[2016,2025,2035,2041,2051,2056,2065,2073,2073,2085,2090,2102,2106,2107,2118,2123,2131,2132,2140,2152,2156,2168,2173,2185,2190,2202,2206,2207,2215,2223,2235,2240,2240,2252,2256,2257,2269,2273,2285,2290,2302,2306,2318,2324],[2536,2553,2558,2559,2565,2567,2574,2585,2590,2602,2607,2618,2623,2626,2635,2640,2640,2648,2657,2668,2673,2673,2686,2690,2690,2702,2706,2707,2719,2724,2735,2740,2752,2757,2769,2785,2790,2802,2807,2819,2823,2835,2840,2840,2852,2859,2860,2869,2873,2886,2890,2902,2907,2919,2923,2941,2949,2957,2969,2971],[3355,3369,3372,3374,3382,3390,3402,3407,3419,3423,3424,3436,3441],[3610,3625,3629,3634,3640,3652,3657,3669,3674,3686,3690,3691,3702,3707],[3963,3968,3975,3982,4007,4018,4024,4036,4040,4041,4053,4057,4057,4070,4074,4086,4091,4103,4107,4119,4124,4136,4140,4153,4157,4157,4170,4174,4174,4186,4190,4191,4203,4207,4208,4217,4224,4236,4241,4253,4257,4269,4274,4282,4291,4291,4303,4307,4316,4324,4325,4336,4340,4341,4349,4350]],"version":"2.0.0"} et le point 3,0.

    Solution

    En comparant l'équation donnée à la forme générale, nous obtenons a=5, b=-2 et c=0, et x1,y1=3,0.

    En rappelant la formule de la distance et en substituant toutes les valeurs correspondantes, nous obtenons ,

    d=ax0+by0+ca2+b2=5(3)+0+052+(-2)2=1529

    Ainsi, la distance entre la ligne 5x-2y=0 et (3,0) est de 1529 unités.

    Exemple de distance entre un point et une ligne

    Trouve la distance entre les deux lignes dont les équations sont données par y=2x-5 et y=2x+3.

    Solution

    Remarque que les deux droites ont la même pente, 2, ce qui implique qu'elles sont parallèles.

    Pour trouver la distance entre les deux lignes, nous pouvons prendre un point sur l'une de ces lignes et utiliser la formule "distance d'un point à une ligne" pour obtenir la distance.

    Trouvons le point sur la première ligne y=2x-5 en substituant x=0 (comme c'est facile à calculer, toute autre valeur de x aurait été tout aussi valable),

    y=2(0)-5=-5

    Ainsi, un point de la ligne est (0,-5). Nous pouvons maintenant utiliser la formule que nous avons dérivée plus tôt pour trouver la distance entre ce point et la ligne y=2x+3.

    d=ax1+by1+ca2+b2

    Mais d'abord, nous réécrivons y=2x+3 sous la forme générale pour obtenir -2x+y-3=0, et donc nous avons a=-2, b=1 et c=-3, que nous substituons dans l'équation ci-dessus pour obtenir d :

    d=0-5-35=85

    Par conséquent, la distance entre le point (0,-5) et la ligne y=2x+3 est de 85 unités.

    Mais rappelle-toi que ce point se trouve sur la ligne y=2x-5 et que ces lignes sont parallèles, la distance entre les deux lignes est donc également de 85 comme décrit précédemment.

    Distance entre un point et une ligne - Principaux enseignements

      • La distance entre un point et une ligne peut être mesurée de plusieurs façons distinctes, mais c'est la distance la plus courte entre eux qui compte.
      • La distance entre un point et une ligne est mesurée par la distance la plus courte entre eux, c'est aussi la même chose que la distance perpendiculaire entre eux.
      • Soit une ligne donnée par ax+by+c=0, alors la distance entre cette ligne et le point x1,y1 est donnée par la formule d=ax1+by1+ca2+b2.
    Questions fréquemment posées en Distance d'un point à une ligne
    Qu'est-ce que la distance d'un point à une ligne en mathématiques ?
    La distance d'un point à une ligne est la longueur du segment perpendiculaire tracé du point à la ligne.
    Comment calculer la distance entre un point et une ligne ?
    La distance se calcule en utilisant la formule de la distance dans un plan cartésien, souvent via la formule \\frac{|Ax + By + C|}{\sqrt{A^2 + B^2}}.
    Pourquoi la distance d'un point à une ligne est-elle importante ?
    Cette distance est essentielle pour déterminer la position relative d'un point par rapport à une ligne et pour résoudre des problèmes de géométrie.
    Quelle est la formule de la distance d'un point à une droite en coordonnées cartésiennes ?
    La formule est \\frac{|Ax + By + C|}{\sqrt{A^2 + B^2}}, où A, B, et C sont les coefficients de l'équation de la ligne Ax + By + C = 0.

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