Question

For the curve $y^n=a^{n-1}x$, the subnormal at any point is constant.

The value of n must be

• A. $2$
• B. $3$
• C. $0$
• D. $1$

Answer 1

The correct option is A

$2$

Explanation for the correct option:

Step 1. Differentiate the given expression with respect to x:

$\begin{array}{rcl}n{y}^{n-1}\frac{dy}{dx}& =& a^{n-1}\\ \frac{dy}{dx}& =& \frac{a^{n-1}}{n{y}^{n-1}}\end{array}$

Step 2. Find the length of subnormal:

Length of the subnormal is given by $\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}}$

Now the subnormal on the curve $y^n=a^{n-1}x$ is

$\begin{array}{rcl}y\frac{dy}{dx}& =& y\frac{a^{n-1}}{n{y}^{n-1}}\\ & =& \frac{a^{n-1}}{n{y}^{n-2}}\end{array}$

Will be a constant therefore, ${\mathrm{y}}^{\mathrm{n}-2}=1^0$

equating powers, we get $\mathrm{n}=2$

Hence, option (A) is the correct answer.