Question

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

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

Answer 1

The correct option is D

$y^3$

Explanation for the correct option:

Step1. Differentiate the given equation with respect to x:

$$\begin{gathered} xy=c^2 \\ \Rightarrow y=\frac{c^2}{x} \\ \Rightarrow \frac{dy}{dx}=\frac{-c^2}{x^2} \end{gathered}$$

Step 2. Subnormal of any curve is given by

Length of the subnormal is given $\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}}$

Now length of subnormal on $\mathrm{xy}={\mathrm{c}}^2$,

$\begin{array}{rcl}y\frac{dy}{dx}& =& y\frac{-c^2}{x^2}\\ & =& y\frac{-c^2}{{\displaystyle \frac{c^4}{y^2}}}\\ & =& \frac{y^3}{c^2}\end{array}$

Therefore, the subnormal varies as $y^3$

Hence, option (D) is correct answer