The locus of point of intersection of tangents at the ends of normal chord of the hyperbola x^2-y^2=a^2 is

Step-by-step solution

Let p(h, k) be the point of intersection of tangents at the ends of a normal chord of the hyperbola, x^2-y^2=a^2 , then the equation of the chord is h x-k y=a^2 ...(i) But its is normal chord. So, its equation must be of the form x +y =2 a ...(ii) Eqs. (i) and (ii) represents the same line h = -k = 2 a a^2 = a 2 h , = -a 2 k ^2 - ^2 =1 array ll & a^2 4 h^2 - b^2 4 k^2 =1 & a^2 (k^2-h^2 )=4 h^2 k^2 array Therefore, the locus of p(h, k) is a^2 (y^2-x^2 )=4 x^2 y^2