Line passing through the point $P (2, 3)$ meets the lines represented by$x^2-2xy-y^2=0$ at the points A and B such that $PA.PB=17$, the equation of the line is

detailed solution

Correct option is B

Let the equation of the line through P(2, 3) making an angle θ with the positive direction of x-axis bex−2cos⁡θ=y−3sin⁡θ.Then the coordinates of any point on this line at a distance r from p are (2+rcos⁡θ,3+rsin⁡θ). If PA=r1 and PB=r2, then r1,r2 are the roots of the equation.(2+rcos⁡θ)2−2(2+rcos⁡θ)(3+rsin⁡θ)−(3+rsin⁡θ)2=0⇒r2(cos⁡2θ−sin⁡2θ)−2r(cos⁡θ+5sin⁡θ)−17=0⇒17=PA⋅PB=r1r2=17cos⁡2θ−sin⁡2θ⇒cos⁡2θ−sin⁡2θ=1 which is satisfied by θ=0 and thusthe equation of the line is y = 3.