Question

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

• A. $n=0$
• B. $n=1$
• C. $n=2$
• D. $n=-3$

Answer 1

The correct option is C $n=2$

We have,

$y^n=x{a}^{n-1}$ ……. (1)

On differentiating both sides w.r.t $x$, we get

$n{y}^{n-1}\frac{dy}{dx}=a^{n-1}$

$\frac{dy}{dx}=\frac{a^{n-1}}{n{y}^{n-1}}$

We know that the length of the subnormal

$=\left|y\frac{dy}{dx}\right|$

Thus,

$=|y\times \frac{a^{n-1}}{n{y}^{n-1}}|$

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

For the subnormal at any point on the curve is constant and independent of $x$ and $y$

Therefore,

$2-n=0$

$n=2$

Hence, this is the answer.