Question

The equation of the trajectories which is orthogonal to the family of curves $\cos y=a{e}^{-x}$ is

• A. $\sin y=c{e}^x$
• B. $\cos y=c{e}^x$
• C. $\sin y=c{e}^{-x}$
• D. None

Answer 1

Answer: (C)

$+\sin y\frac{dy}{dx}=+a{e}^{-x}$
$\sin y\frac{dy}{dx}=\cos y\Rightarrow \frac{dy}{dx}=\cot y$
equation to the orthogonal trajectory will be obtained by solving the differential equation
$-\frac{dy}{dx}=\cot y$
$c-x=\ln (\sin y)\Rightarrow \sin y=c^{-x}]$