Question

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

• A. $1$
• B. $3$
• C. $2$
• D. $0$

Answer 1

The correct option is C $2$

$y^n=a^{n-1}x$
The length of subnormal = $y_1\times {\left(\frac{dy}{dx}\right)}_{(x_1,y_1)}$
$y^n=a^{n-1}x$
$\Rightarrow n{y}^{n-1}\frac{dy}{dx}=a^{n-1}\times 1$
$\Rightarrow \left(\frac{dy}{dx}\right)=\frac{a^{n-1}}{n\times y^{n-1}}$
$\Rightarrow {\left(\frac{dy}{dx}\right)}_{\left(x_{1,y_1}\right)}=\frac{a^{n-1}}{n\times y_1^{n-1}}$
The length of subnormal $=y_1\times {\left(\frac{dy}{dx}\right)}_{(x_1,y_1)}=K$(constant)
$y_1\times \frac{a^{n-1}}{n\times y_1^{n-1}}=K$
$\Rightarrow \frac{1}{n}{a}^{n-1}\times y_1^{1-n+1}=K$
$\Rightarrow n=2$
Hence, option 'C' is correct.