Given curve is \(x^{2 / 3}+y^{2 / 3}=a^{2 / 3}\) ...(i)
Now, let a point on curve \(P\left(a \cos ^3 \theta, a \sin ^3 \theta\right)\)
On differentiating the curve (i) w.r.t \(x\), we get
\(\frac{2}{3} x^{-1 / 3}+\frac{2}{3} y^{-1 / 3} \frac{d y}{d x}=0 \Rightarrow \frac{d y}{d x}=-\left(\frac{y}{x}\right)^{1 / 3}\)
\(\therefore\) Slope of tangent at point \(P\) is
\(m=\left.\frac{d y}{d x}\right|_P=-\tan \theta\)
\(\therefore\) Equation of the tangent of the curve at point \(P\) is
\(y-a \sin ^3 \theta=-\tan \theta\left(x-a \cos ^3 \theta\right) \quad ...(ii)\)
\(\because\) The tangent (ii) meets the axes at \(A\) and \(B\), so \(A(a \cos \theta, 0)\) and \(B(0, a \sin \theta)\)
\(\therefore \quad A B=\sqrt{a^2 \cos ^2 \theta+a^2 \sin ^2 \theta}=a\)
Hence, option (c) is correct.