From any point on the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=1` , tangents are drawn to the hyperbola `(x^2)/(a^2)-(y^2)/(b^2)=2.` The area cut-off by the chord of contact on the asymptotes is equal to `a/2` (b) `a b` (c) `2a b` (d) `4a b`

Options

A. `(ab)/(2)`
B. `ab`
C. `2ab`
D. `4ab`

Explanation

Let `P(x_(1),y_(1))` be a point on the hyperbola
`(x^(2))/(a^(2))-(y^(2))/(b^(2))=1`. Then, `(x_(1)^(2))/(a^(2))-(y_(1)^(2))/(b^(2))=1`
The chord of contact of tangents from `P` to the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=2` is
`(x x_(1))/(a^(2))-(y y_(1))/(b^(2))=2`..........`(i)`
The equations of the asymptotes are
`(x)/(a)-(y)/(b)=0` and `(x)/(a)+(y)/(b)=0`
The points of intersection of `(i)` with the two asymptotes are given by
`x_(1)=(2a)/((x_(1))/(a)-(y_(1))/(b))`, `y_(1)=(2b)/((x_(1))/(a)-(y_(1))/(b))`,
`x_(2)=(2a)/((x_(1))/(a)+(y_(1))/(b))`, `y_(2)=(2b)/((x_(1))/(a)+(y_(1))/(b))`,
`:.` Area of the triangle `=(1)/(2)(x_(1)y_(2)-x_(2)y_(1))=(1)/(2)((4abxx2)/((x_(1)^(2))/(a^(2))-(y_(1)^(2))/(b^(2))))=4ab`