Question

Equation of the curve passing through $(2,1)$ which has constant sub-tangent of length $k$, is:

• A. $k\ln |y|=x-2$
• B. $k{e}^y=x-2$
• C. $k\ln |y|=2-x$
• D. $k{e}^y=2-x$

Answer 1

Answer: (A), (C)

$k\ln |y|=2-x$

We are given that the length of sub-tangent $=\left|\frac{y}{\frac{dy}{dx}}\right|=k$
$\Rightarrow \pm k\frac{dy}{y}=dx$
Integrating both sides we get, $\pm k\ln |y|=x+c$
Given that the curve passes through $(2,1)\Rightarrow 2+c=0$
Hence the equation of such curve is $\pm k\ln |y|=x-2$