Question

If the length of the subnormal at any point on the curve $x{y}^n=a^{n+1}(a\ne 0)$ is constant, then the value of $n$ is

• A. $1$
• B. $-1$
• C. $2$
• D. $-2$

Answer 1

The correct option is D $-2$

$x{y}^n=a^{n+1},a\ne 0$

$y^n+nx{y}^{n-1}\frac{dy}{dx}=0$

i.e. $\frac{dy}{dx}=-\frac{y}{nx}$

$\therefore$ length of subnormal

$=|y\cdot \frac{dy}{dx}|=\left|\frac{y^2}{nx}\right|=\left|\frac{y^2\cdot y^n}{n{a}^{n+1}}\right|$

$=\left|\frac{y^{n+2}}{n{a}^{n+1}}\right|$ is constant only if $n=-2$.