Question

Determine $p$ such that the length of the subtangent and subnormal is equal for the curve $y=e^{px}+px$ at the point $(0,1)$.

Answer 1

For the curve $y=e^{Px}+Px$

length of sub-tangent $=$ length of sub-normal (at $(0,1)$).

Length of sub-tangent$=T{P}^1=\left|\frac{y_1}{m}\right|$

Length of subnormal $=P^{\prime }N=|y_{1}m|$

$m:$ slope of tangent at $(x_1,y_1)$

$y=e^{Px}+Px$

at $(0,1)$

$\frac{dy}{dx}=e^{Px}\left(P\right)+P$

$\frac{dy}{dx}{|}_{(0,1)}=2P$

So, $\left|\frac{y_1}{m}\right|=|y_{1}m|$

$\left|\frac{1}{2P}\right|=|2P|$

$|4P^2|=1$

$P=\pm \frac{1}{2}$.