show that the length of the portion of any tangent line to the astroid
x^{2/3}+y^{2/3}=a^{2/3}
cut off by the coordinate axes is constant
Here are the steps I used to solve the problem:
1.) Use implicit differentiation to find $\displaystyle \frac{dy}{dx}$.
2.) Use the slope from step 1, and the point-slope formula to write the tangent line, using the given implicit relationship to simplify.
3.) Express the tangent line in the two-intercept form $\displaystyle \frac{x}{a}+\frac{y}{b}=1$.
4.) Use the distance formula to find the distance between the two intercepts, again using the given implicit relationship to simplify.
5.) You should find this distance depends only on the given constant a.
I would simplify, and write:
$\displaystyle \frac{dy}{dx}=-\left(\frac{y}{x} \right)^{\frac{1}{3}}$
Now, use a general point, such as $\displaystyle (x_0,y_0)$ and write the tangent line at this point (using the point-slope formula) as:
$\displaystyle y=-\left(\frac{y_0}{x_0} \right)^{\frac{1}{3}}(x-x_0)+y_0$
Now first, write this in the slope-intercept form, and you will find a nice simplification using the original implicit relation.
Then , write the line in the two-intercept form, and compute the distance between the two intercepts.
edit: Once you have the line from the point-slope formula, you could simply compute the intercepts directly from it.