Question

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

The correct option is C $2a^2$

$\frac{dy}{dx}=\frac{{-a}^2}{x^2}$

$x_1{y}_1=a^2$

$y-y_1=\frac{{-a}^2}{x_1^2}(x-x_1)$

$x=0;y-y_1=\frac{{+a}^2}{x_1^2}(0-x_1)$

$x=0;y=y_1+\frac{{+a}^2}{x_1}$

$y=0;{+y}_1=\frac{{+a}^2}{x_1^2}(x-x_1)$

$x=x_1+\frac{x_1^2{y}_1}{a^2}$

Area $=\frac{1}{2}(x_1+\frac{x_1^2{y}_1}{a^2})(y_1+\frac{a^2}{x_1})$

$=\frac{1}{2}(x_1{y}_1+a^2+\frac{{\left(x_1{y}_1\right)}^2}{a^2}+x_1{y}_1)$

$=\frac{1}{2}(a^2+a^2+\frac{a^4}{a^2}+a^2)$

$=\frac{4a^2}{2}=2a^2$