Orthogonal Trajectories of Parabolas

August 19, 2024

Problem

Find the family of curves that are orthogonal to the family of parabolas

$P = {P_{k} : y = k x^{2} | k \in R}$

Solution

Differentiating

$y = k x^{2}$

gives:

$\frac{d y}{d x} = 2 k x$

Since the product of the slopes of perpendicular lines is -1, for the orthogonal curves:

$\frac{d y}{d x} = - \frac{1}{2 k x}$

From the original equation, we have:

$k = \frac{y}{x^{2}}$

Thus,

$\frac{d y}{d x} = - \frac{1}{2 \frac{y}{x^{2}} x} = - \frac{x}{2 y}$

Rearranging gives:

$x d x + 2 y d y = 0$

Integrating both sides yields:

$\frac{1}{2} x^{2} + y^{2} = C$

Letting $C = c^{2}$ and rearranging:

$\frac{x^{2}}{(\sqrt{2} c)^{2}} + \frac{y^{2}}{c^{2}} = 1$

The required family of curves is:

$E = {E_{c} : \frac{x^{2}}{(\sqrt{2} c)^{2}} + \frac{y^{2}}{c^{2}} = 1 | c \in R}$

This represents a family of ellipses with the major axis being $\sqrt{2}$ times the minor axis, with the major axis along the x-axis and the minor axis along the y-axis.

Additional Note

The family of curves orthogonal to a given family of curves is called the orthogonal trajectories.

As a generalization, find the orthogonal trajectories of the family of ellipses where the major axis is $a$ times the minor axis, with the major axis along the x-axis and the minor axis along the y-axis.

$E = {E_{c} : \frac{x^{2}}{(a c)^{2}} + \frac{y^{2}}{c^{2}} = 1 | c \in R}$

Differentiating,

$\frac{x^{2}}{(a c)^{2}} + \frac{y^{2}}{c^{2}} = 1$

gives:

$\frac{2 x}{(a c)^{2}} d x + \frac{2 y}{c^{2}} d y = 0$

$\frac{x}{a^{2}} d x + y d y = 0$

For orthogonal trajectories:

$y d x - \frac{x}{a^{2}} d y = 0$

Separating variables:

$\frac{d y}{y} = a^{2} \frac{d x}{x}$

Integrating:

$\log | y | = a^{2} \log | x | + C$

Letting $k = e^{C}$ :

$y = k | x |^{a^{2}}$

Therefore, the required family of curves is:

$P = {P_{k} : y = k | x |^{a^{2}} | k \in R}$

This represents a family of power functions.