Question

Equation of a family of curve is $\sin y=k{e}^{x^2}$, what is the differential equation of the family of its orthogonal trajectory?

• A. $\frac{dy}{dx}=-\frac{\tan y}{2x}$
• B. $\frac{dy}{dx}=-\frac{\sin y}{2x}$
• C. $\frac{dy}{dx}=-\frac{\cos y}{2x}$
• D. $\frac{dy}{dx}=-\frac{\cot y}{2x}$

Answer 1

Answer: (D)

$\frac{dy}{dx}=-\frac{\cot y}{2x}$

We have, $\sin y=k{e}^{x^2}\dots \left(1\right)$

Differentiating w.r.t. $x$, we get

$\cos y\frac{dy}{dx}=2xk{e}^{x^2}\dots \left(2\right)$

Eliminating $k$, from (2) using (1), we get

$\cos y\frac{dy}{dx}=2x\sin y$

For orthogonal trajectory, substituting $\frac{dy}{dx}$ with $-\frac{dx}{dy}$ in (2), we get

$-\cos y\frac{dx}{dy}=2x\sin y$

$⟹\frac{dy}{dx}=-\frac{\cot y}{2x}$