Question

Find the value of $n$, so that the subnormal at any point on the curve $x{y}^n=a^{n-1}$ may be constant

• A. $n=-2$
• B. $n=-1$
• C. $n=1$
• D. $n=-3$

Answer 1

The correct option is A $n=-2$

Given $x{y}^n=a^{n-1}..\left(1\right)$
On differentiating w.r.t $x$, e get
$y^n+xn{y}^{n-1}\frac{dy}{dx}=0$
$\Rightarrow \frac{dy}{dx}=-\frac{y}{nx}$
Hence length of sub normal is $|y\frac{dy}{dx}|=\frac{y^2}{nx}=\frac{y^{n+2}}{n{a}^{n-1}}$ .....using (1)
Thus for length of sub normal to be independent of point $n+2=0$

$\Rightarrow n=-2$