Single Variable Calculus, Chapter 3, Review Exercises, Section Review Exercises, Problem 92

Prove that the length of the portion of any tangent line to the astroid $x^{\frac{2}{3}} + y^{\rangle2 3} = a^{\frac{2}{3}}$ cut-off by the coordinate axes is constant.

Solution:

Taking the derivative of the equation of astroid implicitly,


$
\begin{equation}
\begin{aligned}

\frac{d}{dx} (x^{\frac{2}{3}}) + \frac{d}{dy} (y^{\frac{2}{3}}) \frac{dy}{dx} =& \frac{d}{dx} (a^{\frac{2}{3}})
\\
\\
\frac{2}{3} x^{\frac{-1}{3}} + \frac{2}{3} y^{\frac{-1}{3}} \frac{dy}{dx} =& 0
\\
\\
\cancel{\frac{2}{3}} y^{\frac{-1}{3}} \frac{dy}{dx} =& -\cancel{\frac{2}{3}} x^{\frac{-1}{3}}
\\
\\
\frac{dy}{dx} =& - \left( \frac{y}{x} \right) ^{\frac{1}{3}}
\end{aligned}
\end{equation}
$


Equation of the tangent at point $(x_1, y_1)$ in astroid is


$
\begin{equation}
\begin{aligned}

y - y_1 =& - \left( \frac{y_1}{x_1} \right) ^{\frac{1}{3}} (x - x_1)
\\
\\
y - y_1 =& x_1 (y_1)^{\frac{1}{3}} + y_1^{\frac{1}{3}}

\end{aligned}
\end{equation}
$