The locus of the point of intersection of the tangents at the end-points of normal chords of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`, is

Options

A. `(a^(6))/(x^(2))+(b^(6))/(y^(2))=(a^(2)+b^(2))^(2)`
B. `(a^(6))/(x^(2))-(b^(6))/(y^(2))=(a^(2)+b^(2))^(2)`
C. `(a^(6))/(x^(2))-(b^(6))/(y^(2))=(a^(2)-b^(2))^(2)`
D. `(a^(6))/(x^(2))+(b^(6))/(y^(2))=(a^(2)-b^(2))^(2)`

Explanation

Let `P(h,k)` be the point of intersection of tangents at the end-points of a normal chord
`ax costheta+bycottheta=(a^(2)+b^(2))`……..`(i)`
The equation of the chord of contact of tangents drawn from `P(h,k)` to the hyperbola is
`(hx)/(a^(2))-(ky)/(b^(2))=1`.........`(ii)`
Clearly, `(i)` and `(ii)` represent the same line.
`:.(a^(3)costheta)/(h)=(-b^(3)cottheta)/(k)=(a^(2)+b^(2))/(1)`
`impliessectheta=(a^(3))/(h(a^(2)+b^(2)))` and `tantheta=-(b^(3))/(k(a^(2)+b^(2)))`
`impliessec^(2)theta-tan^(2)theta=(a^(6))/(h^(2)(a^(2)+b^(2))^(2))-(b^(6))/(k^(2)(a^(2)+b^(2))^(2))`
`implies(a^(6))/(h^(2))-(b^(6))/(k^(2))=(a^(2)+b^(2))^(2)`
Hence, the locus of `P(h,k)` is `(a^(6))/(h^(2))-(b^(6))/(k^(2))=(a^(2)+b^(2))^(2)`