Question

A tangent is drawn at the point M (1, 8) to the curve $y=\sqrt{(5-x^{2/3}{)}^3}.$ Find the length of its segment included between the coordinate axes.

Answer 1

$y={\sqrt{(5-x^{\frac{2}{3}})}}^3$

$\frac{dy}{dx}=\frac{3}{2}(5-x^{\frac{2}{3}}{)}^{\frac{1}{2}}.(\frac{-2}{3}{x}^{\frac{-1}{3}})$

$\begin{array}{c}\frac{dy}{dx}\end{array}|x=1=-2$

For tangent at $M(1,8)\Rightarrow \frac{y-8}{x-1}=-2\Rightarrow 2x+y=10$

$\Rightarrow \frac{x}{5}+\frac{y}{10}$

$\therefore$ Length of the segment $=\sqrt{5^2+{10}^2}=5\sqrt{5}$