Question

For the curve $y^n=x^{n-1}$ if the subnormal at any point is a constant then $n$ is equal to

• A. $1$
• B. $2$
• C. $-2$
• D. $-1$

Answer 1

Answer: (B)

$2$

Given equation of curve:

$y^n=x^{n-1}$

$\Rightarrow y=x^{\frac{n-1}{n}}$

Differentiating both sides, we get

$y^{\prime }=\left(\frac{n-1}{n}\right)x^{-\frac{1}{n}}$

Subnormal length $=y{y}^{\prime }$

$\Rightarrow y{y}^{\prime }=x^{\frac{n-1}{n}}\left(\frac{n-1}{n}\right)x^{-\frac{1}{n}}$

$\Rightarrow y{y}^{\prime }=\left(\frac{n-1}{n}\right)x^{\frac{n-2}{n}}$

It is given that subnormal is constant at any point.

For this to happen, length should not be variable

$⟹n=2$