Question

From any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ tangents are drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=2$. Then area cut-off by the chord of contact on the asymptotes is equal to :

• A. $\frac{a}{2}$
• B. $ab$
• C. $2ab$
• D. $4ab$

Answer 1

The correct option is D $4ab$

Let $P(x_1,y_1)$ be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.
So,$\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}=1$
The chord of contact of tangents from P to the hyperbola is given by
$\frac{x{x}_1}{2a^2}-\frac{y{y}_1}{2b^2}=1$ ......(i)
The equations of the asymptotes are
$\frac{x}{\sqrt{2}a}-\frac{y}{\sqrt{2}b}=0$ ......(ii)
and $\frac{x}{\sqrt{2}a}+\frac{y}{\sqrt{2}b}=0$ ......(iii)
The point of intersection of (i) with the (ii) is given by
$X_1=\frac{2a^{2}b}{b{x}_1-a{y}_1},Y_1=\frac{2{ab}^2}{b{x}_1+a{y}_1}$
The point of intersection of (i) with the (iii) is given by
$X_2=\frac{2a^{2}b}{b{x}_1+a{y}_1},Y_2=\frac{2{ab}^2}{b{x}_1-a{y}_1}$
Area of required triangle $=\frac{1}{2}(X_1{Y}_2-X_2{Y}_1)$
$=\frac{1}{2}\left(\frac{8a^3{b}^3}{b^2{x}_1-a^2{y}_1}\right)=4ab$