If the lines, joining the origin to the points of intersection of the curve 2 x 2 - 2 x y + 3 y 2 + 2 x - y - 1 = 0 and the line x + 2 y = k , are at right angles, then k 2 equals

Step-by-step solution

Equation of line x + 2 y = k Also x + 2 y k = 1     … 1 2 x 2 - 2 x y + 3 y 2 + 2 x - y - 1 = 0     … 2 Homogenise equation 2 with respect to 1 2 x 2 - 2 x y + 3 y 2 + 2 x - y 1 - 1 1 2 = 0 2 x 2 - 2 x y + 3 y 2 + 2 x - y x + 2 y k - 1 x + 2 y k 2 = 0 2 x 2 - 2 x y + 3 y 2 k 2 + 2 x - y x + 2 y k - x 2 + 4 y 2 + 4 x y = 0 x 2 2 k 2 + 2 k - 1 + x y - 2 k 2 + 3 k - 4 + y 2 3 k 2 - 2 k - 4 = 0 Given these lines are perpendicular then Coefficient of x 2 + Coefficient of y 2 = 0 2 k 2 + 2