<s>[INST] 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?
 [/INST] Now, if $x=0$ we obtain $y=q+\sqrt[3]{qp^2}$ and for $y=0$ we obtain $x=p+\sqrt[3]{pq^2}.$
Thus, $A\left(p+\sqrt[3]{pq^2},0\right)$ and $B\left(0,q+\sqrt[3]{qp^2}\right)$ they are intersection points of the tangent with $x$-axis and with $y$- axis respectively.
Id est, $$AB=\sqrt{\left(p+\sqrt[3]{pq^2}\right)^2+\left( q+\sqrt[3]{qp^2}\right)^2}=$$
$$=\sqrt{p^{\frac{2}{3}}\left(p^{\frac{2}{3}}+q^{\frac{2}{3}}\right)^2+q^{\frac{2}{3}}\left(q^{\frac{2}{3}}+p^{\frac{2}{3}}\right)^2}=
p^{\frac{2}{3}}+q^{\frac{2}{3}}=a^{\frac{2}{3}}.$$
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