Question

Find the area of the triangle formed from the coordinate axes and the tangent line to the curve

Answer 1

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

$\begin{aligned} \dfrac{dy}{dx} &= \dfrac{−a ^2}{x^2}\\ \\ x _1y _1&=a ^2\\ ​ y−y _1&​= \dfrac{-a²}{x_1^2}(x−x _1)\\ ​\\ ​ x=0;y−y _1&​= \dfrac{+a²}{x_1^2}(0−x _1)\\ ​\\ x=0;y&=y _1\dfrac{+a²}{x_1^2}\\ ​\\ y=0;+y _1​​&= \dfrac{+a²}{x_1^2}(x−x _1)\\ ​\\ x&=x _1+ \dfrac{x_1^2y_1}{x²}\\ \\ \text{Area }&=\dfrac12 (x_1 +\dfrac{x_1^2y_1}{x²}) ​(y_1+ \dfrac{a ^2}{x _1})\\ \\ &= \dfrac12(x _1​y _1+a ^2​+ a ^2+\dfrac{(x _1y _1​) ^2}{a^2}+x _1​y _1​)\\ \\ &= \dfrac12​(a ^2​+a ^2+ \dfrac{a^4}{a ^2}+a^2)\\ ​&= \dfrac{4a²}{2}\\\\ ​&=2a^2 \end{aligned}$