Question

The value of $p$ such that the length of the sub-tangent and sub-normal is equal for the curve $y=e^{px}+px$ at the point $(0,1)$ is

• A. $\pm \frac{1}{2}$
• B. $\pm \frac{2}{3}$
• C. $\pm 1$
• D. $\pm \frac{1}{3}$

Answer 1

The correct option is A $\pm \frac{1}{2}$

$\frac{dy}{dx}=p{e}^{px}+p$ at point $(0,1)=2p$
Length of sub-tangent $=\left|y_1\frac{dx}{dy}\right|$
Length of sub-normal $=\left|y_1\frac{dy}{dx}\right|$

Given, length of sub-tangent = length of sub-normal
$\Rightarrow \frac{dy}{dx}=\pm 1\Rightarrow 2p=\pm 1$
$\Rightarrow p=\pm \frac{1}{2}$