AP EAPCET 05-Jul-2022 Shift 1 Paper

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