UPSC 2024 Maths Optional Paper 1 Q5a — Step-by-Step Solution

10 marks · Section B

Orthogonal trajectories (cartesian and polar) · ODEs · asked 7× in 14 yrs · Read the full method →

Question

Find the orthogonal trajectories of the family of curves $r=c(\sec\theta+\tan\theta)$ , where $c$ is a parameter.

Differentiate to eliminate $c$ ; apply the polar orthogonality rule $dr/d\theta\to -r^2\,d\theta/dr$ ; integrate.

Step 1 — Differential equation of the given family.

Differentiating $r=c(\sec\theta+\tan\theta)$ :

\frac{dr}{d\theta}=c(\sec\theta\tan\theta+\sec^2\theta)=c\sec\theta(\tan\theta+\sec\theta)=\sec\theta\cdot[c(\sec\theta+\tan\theta)]=r\sec\theta.

Step 2 — Orthogonal-trajectory ODE.

Replace $dr/d\theta$ by $-r^2\,d\theta/dr$ (the polar orthogonality rule):

-r^2\frac{d\theta}{dr}=r\sec\theta\;\Rightarrow\;\frac{dr}{d\theta}=-r\cos\theta.

Step 3 — Solve.

\frac{dr}{r}=-\cos\theta\,d\theta\;\Rightarrow\;\ln|r|=-\sin\theta+\text{const}\;\Rightarrow\;r=k\,e^{-\sin\theta}.

\boxed{\;r=k\,e^{-\sin\theta}.\;}