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

Answer: (C)

$x^{2}y=c^3$
Differentiating w.r.t. $x$, we have
$x^2\frac{dy}{dx}+2xy=0$ or $\frac{dy}{dx}=-\frac{2y}{x}$
Equation of the tangent at $(h,k)$ is
$y-k=-\frac{2k}{h}(x-h)$
$y=0$ gives $x=\frac{3h}{2}=a,$ and $x=0$ gives $y=3k=b$.
Now, $a^{2}b=\frac{9h^2}{4}3k=\frac{27}{4}{h}^{2}k=\frac{27}{4}{c}^3$