设直线ax+by=c=0与圆x²+y²=9相交于两点M,N,若c²=a²+b²则向量OM×向量ON(o为坐
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![设直线ax+by=c=0与圆x²+y²=9相交于两点M,N,若c²=a²+b²则向量OM×向量ON(o为坐](/uploads/image/z/8981206-70-6.jpg?t=%E8%AE%BE%E7%9B%B4%E7%BA%BFax%2Bby%3Dc%3D0%E4%B8%8E%E5%9C%86x%26%23178%3B%2By%26%23178%3B%3D9%E7%9B%B8%E4%BA%A4%E4%BA%8E%E4%B8%A4%E7%82%B9M%2CN%2C%E8%8B%A5c%26%23178%3B%3Da%26%23178%3B%2Bb%26%23178%3B%E5%88%99%E5%90%91%E9%87%8FOM%C3%97%E5%90%91%E9%87%8FON%EF%BC%88o%E4%B8%BA%E5%9D%90)
设直线ax+by=c=0与圆x²+y²=9相交于两点M,N,若c²=a²+b²则向量OM×向量ON(o为坐
设直线ax+by=c=0与圆x²+y²=9相交于两点M,N,若c²=a²+b²则向量OM×向量ON(o为坐
设直线ax+by=c=0与圆x²+y²=9相交于两点M,N,若c²=a²+b²则向量OM×向量ON(o为坐
let
M(x1,y1),N(x2,y2)
ax+by+c=0 (1)
x^2+y^2=9 (2)
sub (1) into (2)
x^2+[(-ax-c)/b]^2 =9
b^2x^2 + a^2x^2+2acx+c^2=9b^2
(b^2+a^2)x^2+2acx+c^2-9b^2=0
x1x2=(c^2-9b^2)/(b^2+a^2)
similarly
[(-by-c)/a]^2 +y^2 =9
(a^2+b^2)y^2+2bcy +c^2 -9a^2 =0
y1y2 = (c^2 -9a^2)/(a^2+b^2)
OM.ON
= x1x2+y1y2
= (c^2-9b^2)/(b^2+a^2) + (c^2 -9a^2)/(a^2+b^2)
=[2c^2 -9(a^2+b^2)]/(a^2+b^2)
=(2-9) = -7