If a variable tangent to the curve ${\mathrm{x}}^2\mathrm{y}={\mathrm{c}}^3$ makes intercepts a, b on  x and y-axes, respectively, then the value of  a²b is 

${\mathrm{x}}^2\mathrm{y}={\mathrm{c}}^3$

Differentiating w.r.t. x, we have

$\begin{array}{l} {\mathrm{x}}^2\frac{\mathrm{dy}}{\mathrm{dx}}+2\mathrm{xy}=0\\ \mathrm{or} \frac{\mathrm{dy}}{\mathrm{dx}}=-\frac{2\mathrm{y}}{\mathrm{x}}\end{array}$

Equation of the tangent at (h, k) is

$\begin{array}{l}\mathrm{y}-\mathrm{k}=-\frac{2\mathrm{k}}{\mathrm{h}}\left(\mathrm{x}-\mathrm{h}\right)\\ \mathrm{y}=0 \mathrm{gives} \mathrm{x}=\frac{3\mathrm{h}}{2}=\mathrm{a}\\ \mathrm{and} \mathrm{x}=0 \mathrm{gives} \mathrm{y}=3\mathrm{k}=\mathrm{b}.\\ \mathrm{Now}, {\mathrm{a}}^2\mathrm{b} =\frac{9{\mathrm{h}}^2}{4}3\mathrm{k}=\frac{27}{4}{\mathrm{h}}^2\mathrm{k}=\frac{27}{4}{\mathrm{c}}^3.\end{array}$