A tangent to the hyperbola x^2 4 - y^2 2 =1 meets x -axis at P and y -axis at Q . Lines PR and QR are drawn such that OPRQ is a rectangle (where O is the origin). Then R lies on :

Step-by-step solution

Equation of the tangent at the point ' ' is aligned & x a - y b =1 & P=(a , 0) and Q=(0,-b ) & Let R be ( h , k ) h =a , k =-b & k h = -b a = -b h a k and & = h a aligned By squaring and adding, b^2 h^2 a^2 k^2 + h^2 a^2 =1 aligned & b^2 k^2 +1= a^2 h^2 & a^2 h^2 - b^2 k^2 =1 aligned Now, given q ^ n of hyperbola is x^2 4 - y^2 2 =1 a^2=4, b^2=2 R lies on a^2 x^2 - b^2 y^2 =1 i.e., 4 x^2 - 2 y^2 =1