Question

The value of $k$ for which the length of the sub tangent to the curve $x{y}^k=c^2$ is constant is?

• A. $0$
• B. $1$
• C. $2$
• D. $-2$

Answer 1

The correct option is A $0$

$x{y}^k=c^2$

$\Rightarrow x.x{y}^{k-1}\frac{dy}{dx}+y^k=0$

$\Rightarrow \left(\frac{x}{y}\right)k{y}^k\frac{dy}{dx}+y^k=0$

$\Rightarrow y^k(\frac{kx}{y}\frac{dy}{dx}+1)=0$

$\therefore y=0$ or $\frac{kx}{y}.\frac{dy}{dx}=-1$

$(\frac{dy}{dx}=\frac{-y}{kx})$

Length of subtangent $=y\frac{dx}{dy}=y.\frac{kx}{-y}=-kx$

Also, length of subtangent = constant (independent of $r$)

$\therefore k=0\left(A\right)$