The locus of the middle points of the normal chords of the rectangular hyperbola  $x^2-y^2=a^2$ is

detailed solution

Correct option is A

Equation of the normal at (asec⁡ θ, atan⁡ θ) to the hyperbola x2−y2=a2 isaxsec⁡ θ+aytan⁡ θ=a2+a2=2a2                                                    (1)Let (h, k) be the middle point of this normal then its equation ishx−ky−a2=h2−k2−a2 T=S1⇒ hx−ky=h2−k                                         (2)Comparing (1) and (2) we gethsec⁡θ=−ktan⁡θ=h2−k22a⇒ sec⁡θ=h2−k22ah,tan⁡θ=h2−k2−2ak So  1=sec2⁡ θ−tan2⁡ θ=h2−k224a21h2−1k2⇒ h2−k23+4a2h2k2=0 Locus of (h, k) is y2−x23=4a2x2y2