Orthogonal trajectories are another application of differential equations. We'll explore more of this in this post.

## What are Orthogonal Trajectories?

Consider two families of curves: circles and lines. If we look at the intersection points of these families, we'll notice that these curves meet at right angles. When two families are perpendicular to each other at meeting points, we say that these are mutually orthogonal curves.

In this example, we say that the family of lines \(y = Kx\) is the orthogonal trajectory of the circular curves with the equation \(x^2 + y^2 = C\) or vice versa.

How can we say that two families are mutually orthogonal to each other? If we recall geometry, lines are said to be perpendicular if the product of their slopes \(m\) is -1. Another way of saying this is that the slopes are negative reciprocals of each other.

Based on this, we can say that two families are orthogonal to each other if the product of their first derivatives is -1. From this, we can find the orthogonal trajectory of any given curve.