Question

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

• A. $2$
• B. $-1$
• C. $\frac{1}{2}$
• D. $0$

Answer 1

Answer: (C)

$\frac{1}{2}$

Given curve $y=a^{1-k}{x}^k$ ....(1)
Let $P(x_1,y_1)$ be a point on the curve.
$\Rightarrow y_1=a^{1-k}{x_1}^k$ .....(2)
Differentiating (1) w.r.t. x, we get
$\frac{dy}{dx}=a^{1-k}k{x}^{k-1}$
$m={\left(\frac{dy}{dx}\right)}_{(x_1,y_1)}=a^{1-k}k{x}_1^{k-1}$ .....(3)
Length of subnormal at point P $=|y_{1}m|$
$=a^{2-2k}k{x_1}^{2k-1}$ (by (2) and (3))
Given , length of subnormal is constant at any point on the curve.So it is independent of $x_1$
$\Rightarrow 2k-1=0$
$\Rightarrow k=\frac{1}{2}$