If the tangents drawn from a point on the hyperbola `x^2-y^2=a^2-b^2` to the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` make angle `alpha" and "beta` with the transverse axis of the hyperbola then

Options

A. `tanalpha-tanbeta=1`
B. `tanalpha+tanbeta=1`
C. `tanalphatanbeta=1`
D. `tanalphatanbeta=-1`

Explanation

Let `P(h,k)` be a point on the hyperbola `x^(2)-y^(2)=a^(2)-b^(2)`. Then,
`h^(2)-k^(2)=a^(2)-b^(2)`………..`(i)`
The equation of a tangent to the ellipse `(x^(2))/(a^(2))+(y^(2))/(b^(2))=1` is
`y=mx+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`
Let `m_(1)` and `m_(2)` be the roots of this equation. Then,
`m_(1)m_(2)=(k^(2)-b^(2))/(h^(2)-a^(2))=1` [Using `(i)`]
`implies tan alpha tanbeta=1`