Orthogonal Trajectories of Parabolas

Problem

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

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

Solution

Differentiating

$$ y = k x^2 $$

gives:

$$ \frac{dy}{dx} = 2kx $$

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

$$ \frac{dy}{dx} = -\frac{1}{2kx} $$

From the original equation, we have:

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

Thus,

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

Rearranging gives:

$$ xdx + 2ydy = 0 $$

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:

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

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.

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

Differentiating,

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

gives:

$$ \frac{2x}{(ac)^2}dx + \frac{2y}{c^2}dy = 0 $$

$$ \frac{x}{a^2}dx + ydy = 0 $$

For orthogonal trajectories:

$$ ydx - \frac{x}{a^2}dy = 0 $$

Separating variables:

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

Integrating:

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

Letting $k = e^C$:

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

Therefore, the required family of curves is:

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

This represents a family of power functions.