Question

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

• A. $1/2$
• B. $1$
• C. $2$
• D. $-2$

Answer 1

The correct option is D $-2$

given curve $-x{y}^n=a^{n+1}...\left(1\right)$

differentiate $e{q}^n\left(1\right)$

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

$\frac{dy}{dx}=\frac{-y}{nx}$

normal-

$y\frac{dy}{dx}=c$

$y.\left(\frac{-y}{nx}\right)=c\Rightarrow \frac{-y^2}{nx}=c$

length of subnormal $=\frac{-y^2}{nx}=\frac{-y^2}{a^{n+1}}.y^2$

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

is constant if f $n+2=0$

So, option (D) is right