已知圆x的平方+y的平方+x-6y+m=0与直线x+2y-3=0相交于p.q两点,o为原点,若op垂直oq,求实数的m值.
人气:469 ℃ 时间:2019-09-01 10:49:28
解答
设P(x1,y1),Q(x2,y2)
根据OP⊥OQ:两直线斜率夹角90°,
y1×y2 + x1×x2 = 0 ①
将直线方程代入圆方程(置换掉x):
(3-2y)^2 + y^2 + (3-2y) - 6y + m = 0
即 5y^2 - 20y + 12 + m = 0 有两实根y1、y2
y1 × y2 = (m+12)/5 ②
将直线方程代入圆方程(置换掉y):
x^2 + [(3-x)/2]^2 + x - 6[(3-x)/2] + m = 0
即 5x^2 + 10x -27 + 4m = 0 有两实根x1、x2
x1 × x2 = (4m-27)/5 ③
利用②、③代入①式得到:
(4m-27)/5 + (m+12)/5 =0
m=3
初始的圆方程为:
(x-1/2)^2 + (y-3)^2 = 9 + 1/4 - m = 25/4 ④
圆心O坐标为(-1/2,3),半径为5/2
直线方程:x + 2y - 3 = 0 ⑤
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