Practicing Success
The area of a triangle formed by a tangent to the curve $2 x y=a^2$ and the coordinate axes, is |
$2 a^2$ $a^2$ $3 a^2$ none of these |
$a^2$ |
Let $P(x_1, y_1)$ be a point on the curve $2 x y=a^2$ .....(i) Then, $2 x_1 y_1=a^2$ .....(ii) Now, $2 x y=a^2 \Rightarrow 2\left(x \frac{d y}{d x}+y\right)=0 \Rightarrow \frac{d y}{d x}=-\frac{y}{x} \Rightarrow\left(\frac{d y}{d x}\right)_P=-\frac{y_1}{x_1}$ The equation of the tangent to (i) at $P(x_1, y_1)$ is $y-y_1=-\frac{y_1}{x_1}\left(x-x_1\right)$ $\Rightarrow x_1 y-x_1 y_1=-x y_1+x_1 y_1$ $\Rightarrow x y_1+y x_1=2 x_1 y_1 \Rightarrow x y_1+y x_1=a^2$ [Using (ii)] This tangent meets the coordinate axes at $A\left(a^2 / y_1, 0\right)$ and $B\left(0, a^2 / x_1\right)$ ∴ Area of $\triangle O A B=\frac{1}{2} \times O A \times O B$ ⇒ Area of $\triangle OAB=\frac{1}{2} \times \frac{a^2}{y_1} \times \frac{a^2}{x_1}=\frac{a^4}{2 x_1 y_1}=a^2$ [Using (i)] |