The locus of the point of intersection of the tangents at the endpoints of normal chords of the hyperbola x^2 a^2 - y^2 b^2 =1 is

Step-by-step solution

Let p(h, k) be point of intersection of tangents at end points of normal chord Standard equation of normal chord of hyperbola a x +b y =a^2+b^2 ...(i) for p(h, k) , equation of chord of contact of tangents to hyperbola is h x a^2 - k y b^2 =1 ...(ii) (i) and (ii) represent some line a ( h a^2 ) = b ( -k b^2 ) = a^2+b^2 1 a^3 h = -b^3 k = a^2+b^2 1 aligned & = h (a^2+b^2 ) a^3 & = a^3 h (a^2+b^2 ) aligned aligned & and = -k (a^2+b^2 ) b^3 & = -b^3 k (a^2+b^2 ) aligned ^2 - ^2 =1 aligned & a^6 h^2 (a^2+b^2 )^2 - b^6