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.
Find the Orthogonal trajectories of xy = c
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.