Prove that the locus of the point of intersection of the tangents at the ends of the normal chords of the hyperbola `x^(2)-y^(2)=a^(2)" is " a^(2)(y^(2)-x^(2))=4x^(2)y^(2)`.

Options

A. `y^(4)-x^(4)=4a^(2)x^(2)y^(2)`
B. `y^(2)-x^(2)=4a^(2)x^(2)y^(2)`
C. `a^(2)(y^(2)-x^(2))=4x^(2)y^(2)`
D. `y^(2)+x^(2)=4a^(2)x^(2)y^(2)`

Explanation

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
`hx-ky=a^(2)`………`(i)`
But, it is a normal chord. So, its equation must be of the form
`x cos theta+ycot theta=2a`
Equation `(i)` and `(ii)` represent the same line.
`:. (costheta)/(h)=(cotheta)/(-k)=(2a)/(a^(2))`
`impliessectheta=(a)/(2h)` and `tantheta=-(a)/(2k)`
`:.sec^(2)theta-tan^(2)theta=1implies(a^(2))/(4h^(2))-(a^(2))/(4k^(2))=1impliesa^(2)(k^(2)-h^(2))=4h^(2)k^(2)`
Hence, the locus of `P(h,k)` is `a^(2)(y^(2)-x^(2))=4x^(2)y^(2)`