Let p(h,k) be the point of intersection of tangents at the ends of a normal chord of the hyperbola, x2−y2=a2, then the equation of the chord is hx−ky=a2 .....(i) But its is normal chord. So; its equation must be of the form x‌cos‌θ+y‌cot‌θ=2a.....(ii) Eqs. (i) and (ii) represents the same line ∴‌‌
cos‌θ
h
=‌
cot‌θ
−k
=‌
2a
a2
‌secθ=‌
a
2h′
,tan‌θ=‌
−a
2k
∴‌sec2θ−tan2θ=1 ⇒‌‌‌
a2
4h2
−‌
b2
4k2
=1 ⇒‌‌a2(k2−h2)=4h2k2 Therefore, the locus of p(h,k) is a2(y2−x2)=4x2y2