# Orthogonal Trajectory with Solved Problems

## Definition of orthogonal trajectory

Let us consider the two families of curves with the property: every member of either family cuts each member of the other family at right angles. Such two families of curves are called orthogonal trajectories of each other.

## Method of Finding orthogonal trajectories

• For cartesian form: replace $\frac{dy}{dx}$ by $-\frac{dx}{dy}$
• For polar form: replace $\frac{dr}{d\theta}$ by $-r^2\frac{d\theta}{dr}$
• For ω-trajectories, replace $\frac{dy}{dx}$ by $\dfrac{\frac{dy}{dx} -\tan \omega}{1+\frac{dy}{dx} \tan \omega}$.

Question1: Find the orthogonal trajectories of the curves xy=a2.

xy=a2

Differentiating both sides w.r.t x,

y+x$\frac{dy}{dx}$ =0

Now, replace $\frac{dy}{dx}$ by $-\frac{dx}{dy}$. So we get that

y – x$\frac{dx}{dy}$ =0

⇒ xdx – ydy =0

Integrating, x2-y2=c2.

This is the orthogonal trajectories of the family of curves xy=a2.

Question2: Find the orthogonal trajectories of the straight lines y=mx.

y=mx

Differentiating both sides w.r.t x,

$\frac{dy}{dx}$ =m = y/x

Replacing $\frac{dy}{dx}$ by $-\frac{dx}{dy}$, we get that

$-\frac{dx}{dy}$ =y/x

⇒ xdx + ydy =0

Integrating, x2+y2=c2.

This is the orthogonal trajectories of the family of straight lines y=mx.

Question3: Find the orthogonal trajectories of the curves y2=4ax.

y2=4ax

Differentiating both sides w.r.t x,

2y$\frac{dy}{dx}$ =4a = y2/4a

Replace $\frac{dy}{dx}$ by $-\frac{dx}{dy}$. Thus,

$-2y\frac{dx}{dy}=\frac{y^2}{x}$

⇒ y2dy + 2xydx =0

⇒ ydy + 2xdx =0

Integrating, x2+y2/2=c. This is the desired orthogonal trajectories of y2=4ax that defines a family of concentric ellipses.

Question4: Find the orthogonal trajectories of the curves x2/3+y2/3=a2/3, where a is a parameter.

x2/3+y2/3=a2/3

Differentiating both sides w.r.t x,

2/3 x-1/3 + 2/3 y-1/3=0

⇒ dy/dx = -(y/x)1/3

Replacing $\frac{dy}{dx}$ by $-\frac{dx}{dy}$, we obtain that

dx/dy = (y/x)1/3

⇒ x1/3dx = y1/3dy

Integrating, 3/4 x4/3 – 3/4 y4/3 = 3/4 c4/3

⇒ x4/3 – y4/3 = c4/3

This is the desired orthogonal trajectories of x2/3+y2/3=a2/3.

Cauchy’s Criterion of Series

Abel’s Theorem of Series Convergence

Beta and Gamma Functions