Question

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

• A. $27c^3$
• B. $\frac{4}{27}{c}^3$
• C. $\frac{27}{4}{c}^3$
• D. $\frac{4}{9}{c}^3$

Answer 1

The correct option is C $\frac{27}{4}{c}^3$

Given, $x^{2}y=c^3$
Differentiating w.r.t. $x$, we get
$x^2\frac{dy}{dx}+2xy=0\Rightarrow \frac{dy}{dx}=-\frac{2y}{x}$
Equation of the tangent at $(h,k)$ is $y-k=-\frac{2k}{h}(x-h)$
At $y=0\Rightarrow x=\frac{3h}{2}=a$
At $x=0\Rightarrow y=3k=b$
$\therefore a^{2}b=\frac{9h^2}{4}\cdot 3k=\frac{27}{4}{h}^{2}k=\frac{27}{4}{c}^3$