# Trajectories

## Trajectories

Trajectories:  We are given the family of plane curve F (x, y, a) = 0 depending on a single parameter a.

A curve making at each of its points a fixed angle α with the curve of the family passing through that point is called as isogonal trajectory of that family. If in particular α = π/2, then it is called an Orthogonal Trajectory.

Finding Orthogonal Trajectories:

The differential Equation of the given family of curve. Let it be of the form F (x, y, y’) = 0

The differential equation of the orthogonal trajectories is of the form F (x, y, -1/y’) = 0 and its solution φ₁ (x, y, C) = 0 gives the family of orthogonal trajectories.

Example:

Find the Orthogonal trajectories of xy = c

Solution:

Given that xy = c

Differentiating with respect to ‘x’

d/dx(x). y + d/dx(y). x = 0

y + dy/dx x = 0

dy / dx = -y/x

replace dy/dx by -dx/dy

-dx / dy = -y/x

dx /dy = y/x

by variable separable method

x dx = y dy

Integrating on both sides

∫ x dx = ∫ y dy

x²/2 = y²/2 + c

x² – y² = 2c

x² – y² = C

This is the family of the required orthogonal trajectories.