proves of parametric curves via parametric equations

Question

Hi could anyone help me with this problem.

An astroid is given by the equation $$x^{2/3} + y^{2/3} = 1.$$ Prove via parametric equations that the length of a piece of a tangent line between the coordinate axes is constant.

First I drew out the curve which is a hypocycloid, then changed the above equation to a parametric form by setting $$ \begin{align*} x(a) &= \cos^3 a, \\ y(a) &= \sin^3 a. \end{align*}$$

Then I differentiate it to form $(x'(a),y'(a))$ to get the tangent vector. I then get the equation of the tangent line in the form of $$(x(a),y(a))+t(x'(a),y'(a))$$ but I'm unsure of how to proceed from here.

Answer 1

At the point with parameter $t$ we have $$x=\cos^3t\ ,\quad y=\sin^3t\ ,\quad\frac{dy}{dx}=\frac{3\sin^2t\cos t}{-3\cos^2t\sin t}=-\tan t$$ and so the tangent has equation $$y-\sin^3t=-(\tan t)(x-\cos^3t)\ .$$ It cuts the $x$ and $y$ axes at $$x=\frac{\sin^3t}{\tan t}+\cos^3t=\cos t\quad\hbox{and}\quad y=\cos^3t\tan t+\sin^3t=\sin t$$ respectively, and the distance between these intercepts is $$\sqrt{\cos^2t+\sin^2t}=1\ ,$$ independent of $t$.

Answer 2

Once you have the equation of the tangent line for a particular value of the parameter, then it is easy to find the $x$- and $y$-intercepts. If the equation of the line has the form $$x/a + y/b = 1$$, then obviously the $x$-intercept is $(a,0)$ and the $y$-intercept is $(0,b)$. Then the length of the tangent segment between coordinate axes is simply $\sqrt{a^2 + b^2}$.