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.