Mathematics - Application of Derivative Question with Solution | TestHub

Since, $AP:BP=3:1$ where, equation of tangent is $y-y_1=f^{\prime }(x)\left(x-x_1\right)$ $$\begin{array}{rl}\therefore & A\left(x_1-\frac{y_1}{f^{\prime }\left(x_1\right)},0\right)\text{ and }B\left(0,y_1-x_1{f}^{\prime }(x)\right)\\ & AP:BP=3:1\Rightarrow \frac{dy}{dx}=\frac{y}{-3x}\\ \therefore & 3x\frac{dy}{dx}+y=0\text{ or }\frac{dy}{y}=-\frac{1}{3x}dx,\end{array}$$ On integrating both sides, we get $$\begin{array}{lrlr}\Rightarrow & x{y}^3& =1& \\ & \therefore \text{ At }& x& =\frac{1}{8}\text{ and }y=2\end{array}$$