If the lines joining the origin to the points of intersection of 2 x+3 y=k and 3 x^2-x y+3 y^2+2 x-3 y-4=0 are at right angles, then

Step-by-step solution

Given line 2 x+3 y=k 2 x+3 y k =1 ...(i) Also given, 3 x^2-x y+3 y^2+2 x-3 y-4=0 Now homogenising the above equation aligned & 3 x^2-x y+3 y^2+2 x(1)-3 y(1)-4(1)^2=0 & 3 x^2-x y+3 y^2+2 x ( 2 x+3 y k )-3 y ( 2 x+3 y k ) & -4 ( 2 x+3 y k )^2=0 & 3 x^2-x y+3 y^2+ (4 x^2+6 x y ) k - (6 x y+9 y^2 ) k & - 4 (4 x^2+9 y^2+12 x y ) k^2 =0 & 3 x^2 k^2-x y k^2+3 y^2 k^2+4 x^2 k+6 x y k-6 x y k & -9 y^2 k-16 x^2-36 y^2-48 x y k^2 =0 & aligned x^2 (3 k^2+4 k-16 )+y^2 (3 k^2-9 k-36 )+x y (-k^2-48 )=0 The above equation represen