Find the length of the tangent to the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ which is intercepted between the axes. 
Find the length of the tangent to the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ which is intercepted between the axes.

$x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}\implies \frac{dy}{dx}={(\frac{-y}{x})}^{\frac{1}{3}}$.
Slope at $(p,q)$ will be ${(\frac{-q}{p})}^{\frac{1}{3}}$.
So equation of tangent at $(p,q);$ $y-q={(\frac{-q}{p})}^{\frac{1}{3}}(x-p)$.
How to find the length of tangent intercepted in between the axes from here?

Any point on $$x^{2/n}+y^{2/n}=a^{2/n}\ \  \ \ (1)$$ can be chosen as $(a\cos^nt,a\sin^nt)$
Differentiating $(1)$ wrt $x,$  $$\dfrac{dy}{dx}=-\dfrac{x^{(2-n)/n}}{y^{(2-n)/n}}$$
$$\dfrac{dy}{dx}_{\text{ at }(a\cos^nt,a\sin^nt)}=-\dfrac{x^{(2-n)/n}}{y^{(2-n)/n}}=-\dfrac{\cos^{2-n}t}{\sin^{2-n}t}$$
So, the equation of the tangent at $(a\cos^nt,a\sin^nt)$ will be $$\dfrac{y-a\sin^nt}{x-a\cos^nt}=-\dfrac{\cos^2t\sin^nt}{\sin^2t\cos^nt}$$
$$\iff x\cos^2t\sin^nt+y\sin^2t\cos^nt=a\cos^nt\sin^nt$$
$$\iff\dfrac x{\cos^{n-2}t}+\dfrac y{\sin^{n-2}t}=1$$
Here $n=3$
Can you take it from here?
See also: intercept form 
