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. `a//2`
B. `ab`
C. `2ab`
D. `4ab`

Answer

D

Explanation

Let `P(x_(1),y_(1))` be a point on the hyperbola
`(x^(2))/(a^(2))-(y^(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 given by
`(x x_(1))/(a^(2))-(yy_(1))/(b^(2))=2" "(1)`
The equation of the asymptotes are
`(x)/(a)-(y)/(b)=0`
`and (x)/(a)+(y)/(b)=0`
the points of intersection of (1) with the two asymptotes are given by
`x_(1)=(2)/((x_(1)//a)-(y_(1)//b)),y_(1)=(2b)/((x_(1)//a)-(y_(1)//b))`
`x_(2)=(2)/((x_(1)//a)-(y_(1)//b)),y_(2)=(-2b)/((x_(1)//a)-(y_(1)//b))`
`"Area of the said 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`