Question

The subnormal at any point on the curve $x{y}^n=a^{n+1}$ is constant for :

• A. $n=0$
• B. $n=1$
• C. $n=-2$
• D. $n=2$

Answer 1

The correct option is C $n=-2$

$x{y}^n=a^{n+1}$

eqn of subnormal at any point on $x{y}^n=a^{n+1}$

is constant

$i.e,y\frac{dy}{dx}=k$

When $n=-2.$

$x{y}^{-2}=a^{-1}=\frac{1}{a}$

$\Rightarrow \frac{d}{dx}\left(x{y}^{-2}\right)=0$

$\Rightarrow \frac{1}{y^2}+x\frac{-2}{y^3}\frac{dy}{dx}=0$

$\Rightarrow y^2+2x(-y)\frac{d}{dx}=0$

$\Rightarrow y\frac{dy}{dx}=\frac{y^2}{2x}=\frac{ax}{2x}=\frac{a}{2}$

$\therefore y\frac{dy}{dx}=\frac{a}{2}$, constant at any point.

Hence subnormal is constant at any point form

$n=-2$