Question

If a variable tangent to the curve $x^{2}y=c^3$ makes intercepts $a,b$ on $x$ and $y-$axis 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 $\frac{27}{4}{c}^3$

$x^{2}y=c^3$
$\Rightarrow x^2\frac{dy}{dx}+2xy=0$
$\Rightarrow \frac{dy}{dx}=-\frac{2y}{x}$
Equation of tangent at $(x,y)$
$Y-y=-\frac{2y}{x}(X-x)$
$Y=0,$ gives, $X=\frac{3x}{2}=a$
and $X=0,$ gives, $Y=3y=b$
Now,
$a^{2}b=\frac{9x^2}{4}\times 3y$
$=\frac{27}{4}{x}^{2}y=\frac{27}{4}{c}^3$