If the line x+2 y=k intersects the curve x^2-x y+y^2+3 x+3 y-2=0 at two points A and B and if O is the origin, then the condition for AOB =90^ is

Step-by-step solution

We have, x^2-x y+y^2+3 x+3 y-2=0 and x+2 y=k x+2 y k =1 By homogeneous of Eq. (i), we get aligned x^2-x y+y^2+3 x & ( x+2 y k ) & +3 y ( x+2 y k )-2 ( x+2 y k )^2=0 aligned aligned k^2 x^2-k^2 x y+k^2 y^2+3 k x^2+6 k x y+3 k x y+6 k y^2 & & -2 x^2-8 x y-8 y^2=0 x^2 (k^2+3 k-2 )- (k^2-9 k+8 ) & x y+ (k^2+6 k-8 ) y^2 & =0 aligned Since, A O B=90^ aligned & k^2+3 k-2+k^2+6 k-8=0 & 2 k^2+9 k-10=0 aligned