设直线ax+by=c=0与圆x²+y²=9相交于两点M,N,若c²=a²+b&#

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