Find the value of $a^2b$ . If a variable tangent to the curve $x^2y = c^3$ makes intercepts $a$ and $b$ on the $x$ and $y$ axis respectively then find the value of $a^2b$. 

The tangent in the point $(x_0,\frac{c^3}{x_0^2})$ is $y'=-\frac{2c^3}{x^3}$ so the tangent lines crossing each point of the function are:
$$
y-\frac{c^3}{x_0^2} = -\frac{2c^3}{x_0^3}(x-x_0)
$$
$a$ is where tangent line crosses $x$-axis i.e. $y=0$ , and $b$ is where $x=0$.
$$
\left.
\begin{array}{cc}
y=0 \Rightarrow a &= \frac32 x_0\\
x=0 \Rightarrow b &= 3\frac{c^3}{x_0^2}
\end{array}
\right\}
\Rightarrow a^2b = \frac{27}4c^3
$$
