Question

The area of triangle formed by tangent to the hyperbola $2xy=a^2$ and coordinates axes is -

• A. $a^2$
• B. $2a^2$
• C. $\frac{a^2}{2}$
• D. $\frac{3a^2}{2}$

Answer 1

The correct option is A $a^2$

Given curve is, $2xy=a^2$
Differentiating w.r.t $x$
$2y+2x\frac{dy}{dx}=0\Rightarrow \frac{dy}{dx}=-\frac{y}{x}$
Thus equation of tangent to this curve at any point $P(x_1,y_1)$ is given by,
$(y-y_1)=-\frac{y_1}{x_1}(x-x_1)$
The intercepts on the axis are, $2x_1$ and $2y_1$
Hence are of triangle formed by tangent at P on the coordinate axes is,
$\Delta =\frac{1}{2}\times 2x_1\times 2y_1=2x_1{y}_1=a^2$ since point P also lie on the given curve $(2x_1{y}_1=a^2)$
Hence, option 'A' is correct.