The point of intersection of two tangents to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` , the product of whose slopes is `c^2,` lies on the curve: `y^2-b^2=c^2(x^2+a^2)` (B) `y^2+a^2=c^2(x^2-b^2)` (C) `y^2+b^2=c^2(x^2-a^2)` (D) `y^2-a^2=c^2(x^2+b^2)`

Options

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

Explanation

Let `P(h,k)` be the point of intersection of two tangents to the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`.
The equation of any tangent to the hyperbola is
`y=mx+sqrt(a^(2)m^(2)-b^(2))`
If it passes through `P(h,k)` then
`k=mh+sqrt(a^(2)+m^(2)-b^(2))`
`implies(k-mh)^(2)=a^(2)m^(2)-b^(2)`
`impliesm^(2)(h^(2)-a^(2))-2mhk+(k^(2)+b^(2))=0`.........`(i)`
Let `m_(1)` and `m_(2)` be the slopes of the tangents passing through `P`. Then, `m_(1)`, `m_(2)` are the roots of the equation `(i)`
`:.m_(1)m_(2)=(k^(2)+b^(2))/(h^(2)-a^(2))`
`impliesc^(2)=(k^(2)+b^(2))/(h^(2)-a^(2))implies(h^(2)-a^(2))c^(2)=k^(2)+b^(2)` [` :' m_(1)m_(2)=c^(2)`]
Hence, `P(h,k)` lies on `(x^(2)-a^(2))c^(2)=y^(2)+b^(2)`.