Question

The area of the triangle formed by the coordinates axes and a tangent to the curve $xy=a^2$ at the point $(x_1,y_1)$ on it is ?

• A. $\frac{a^2-x_1}{y_1}$
• B. $\frac{a^2-y_1}{x_1}$
• C. $2a^2$
• D. $4a^2$

Answer 1

Answer: (C)

$2a^2$

Given $\left(x_1{y}_1\right)$ point on curve

i.e. $x_1{y}_1=a^2$

let tangentbe $(y-y_1)=m(x-x_1)$

$m=\frac{dy}{dx}\Rightarrow$ differentiation of curve at $\left(x_1{y}_1\right)$ give slope of line.

$\frac{d\left(xy\right)}{dx}=a^2$

$x\frac{dy}{dx}+y.1=0\Rightarrow \frac{dy}{dx}=\frac{-y}{x}=\frac{-y_1}{x_1}$

tanget equation $=(y-y_1)=\frac{-y}{x}=\frac{-y_1}{x_1}$

$At=y=0\Rightarrow x=2x_1$

$x=0\Rightarrow y=2y_1$

Area $=\frac{1}{2}\times 2x_1\times 2y_1=2x_1{y}_1=2a^2$