Question

For the curve $xy=c^2$ the subnormal at any point varies as

• A. $x^3$
• B. $x^2$
• C. $y^3$
• D. $\infty$

Answer 1

Answer: (C)

$y^3$

The subnormal is given by $y\frac{dy}{dx}$

Here, $xy=c^2$

So, $y=\frac{c^2}{x}$

$\frac{dy}{dx}=\frac{-c^2}{x^2}$

$\therefore y\frac{dy}{dx}=\frac{c^2}{x}\times \frac{-c^2}{x^2}=\frac{-c^3}{x^4}$

$=\frac{-x^2{y}^2}{x^3}=\frac{-y^2}{x}=\frac{-y^3}{c^2}$

Thus, it varies as $y^3$