## Orthogonal Trajectories

Orthogonal trajectories refer to a family of curves that intersect another family of curves at right angles. In other words, if one curve in the first family intersects a curve in the second family, the tangent lines at the point of intersection are perpendicular.

To find the orthogonal trajectories of a given family of curves, we need to determine the differential equation that defines the second family of curves. Then, we solve the differential equation to obtain the curves that intersect the original family of curves at right angles.

For example, let’s consider the family of curves given by the differential equation y=f(x), and m=dy/dx represents the slope of the curve. To find the orthogonal trajectories, we need to determine the differential equation that defines the family of curves that intersect y = f(x) at right angles.

## Method of finding the orthogonal trajectories

The method to find orthogonal trajectories for a family of curves remains the same, regardless of whether they are given in Cartesian or polar form. However, the equations for the orthogonal trajectories may differ depending on the form of the original family of curves.

### Case (i) Cartesian family f(x,y,c)=0

- Consider the family of curves f(x,y,c)=0.
- We differentiate the equation with respect ‘x’ and obtain dy/dx to eliminate the parameter ‘c’.

Replace dy/dx by -dx/dy to obtain a new differential equation. - Solve the new differential equation to obtain the orthogonal trajectories of the given family of curves.

### Case (ii) Polar family f(r,θ,c)=0

- Consider the family of curves f(r,θ,c)=0.
- Take logarithm on both sides of the equation, then differentiate the equation with respect to ‘θ’ and obtain dr/dθ to eliminate the parameter ‘c’.
- Replace dr/dθ by -r^2 dθ/dr to obtain a new differential equation.
- Solve the new differential equation to obtain the orthogonal trajectories of the given family of curves.

## Topic

Applications of Differential Equations

## Problems of Differential Equations

- Find the orthogonal trajectories of the family of curves x^2/a^2 +y^2/ (b^2+λ) =1, where λ is the parameter
- Find the orthogonal trajectories of the family r=a(1+sin(theta))
- Find the orthogonal trajectories of the family 2a/r=1-cos(theta)
- Find the Orthogonal trajectories of the family of curves r^n=a^n cos(n.theta)
- Using the concept of orthogonal trajectories show that the family of curves r=a (sin(theta)+cos(theta)) and r=b (sin(theta)-cos(theta)) intersect each other orthogonally
- Show that the family of parabolas y^2=4a (x+a) is self-orthogonal
- Find the Orthogonal trajectories of the family r^n cos(n.theta) = a^n