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