Let P be any point on the curve $x^{2/3}+y^{2/3}=a^{2/3}$. Then the length of the segment of the tangent between the coordinate axes is of the length

a

Let the coordinates of P be $\left(x_1, y_1\right)$.

We have,

$x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$

$\Rightarrow \frac{2}{3} x^{-1 / 3}+\frac{2}{3} y^{-1 / 3} \frac{d y}{d x}=0$

$\Rightarrow \frac{d y}{d x}=-\frac{y^{1 / 3}}{x^{1 / 3}} \Rightarrow\left(\frac{d y}{d x}\right)_{\left(x_1, y_1\right)}=-\frac{y_1{ }^{1 / 3}}{x_1{ }^{1 / 3}}$

The equation of the tangent at $\left(x_1, y_1\right)$ to the given curve is

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

$\Rightarrow \frac{y-y_1}{y_1{ }^{1 / 3}}=-\frac{x-x_1}{x_1{ }^{1 / 3}}$

$\Rightarrow x x_1^{-1 / 3}+y y_1{ }^{-1 / 3}=x_1{ }^{2 / 3}+y_1{ }^{2 / 3}$

$\Rightarrow x x_1^{-1 / 3}+y y_1^{-1 / 3}=a^{2 / 3}$              $\left[\begin{array}{l}∵\left(x_1, y_1\right) \text { lies on (i)} \\ ∴ x_1{ }^{2 / 3}+y_1{ }^{2 / 3}=a^{2 / 3}\end{array}\right]$

This meets the coordinate axes at $A\left(a^{2 / 3} x_1{ }^{1 / 3}, 0\right)$ and $B\left(0, a^{2 / 3} y_1^{1 / 3}\right)$

∴ AB = $\sqrt{a^{4 / 3} x_1^{2 / 3}+a^{4 / 3} y_1^{2 / 3}}=\sqrt{a^{4 / 3}\left(x_1^{2 / 3}+y_1^{2 / 3}\right)}=a$