Orthogonal Trajectories/Parabolas Tangent to X Axis

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Theorem

Consider the one-parameter family of curves of parabolas which are tangent to the $x$-axis at the origin:

$(1): \quad y = c x^2$


Its family of orthogonal trajectories is given by the equation:

$x^2 + 2 y^2 = c$


ParabolasTangentAxisOrthogonalTrajectories.png


Proof

Differentiating $(1)$ with respect to $x$ gives:

$x \dfrac {\mathrm d y} {\mathrm d x} + y = 0$
\((2):\quad\) \(\displaystyle \frac {\mathrm d y} {\mathrm d x}\) \(=\) \(\displaystyle 2 c x\)
\(\displaystyle \implies \ \ \) \(\displaystyle \frac {\mathrm d y} {\mathrm d x}\) \(=\) \(\displaystyle - \frac {2 y} x\) eliminating $c$ between $(1)$ and $(2)$


Thus from Orthogonal Trajectories of One-Parameter Family of Curves, the family of orthogonal trajectories is given by:

$\dfrac {\mathrm d y} {\mathrm d x} = -\dfrac x {2 y}$


So:

\(\displaystyle \frac {\mathrm d y} {\mathrm d x}\) \(=\) \(\displaystyle -\dfrac x {2 y}\)
\(\displaystyle \implies \ \ \) \(\displaystyle \int x \, \mathrm d x\) \(=\) \(\displaystyle -2 \int y \, \mathrm d y\) Separation of Variables
\(\displaystyle \implies \ \ \) \(\displaystyle x^2\) \(=\) \(\displaystyle -2 y^2 + c\)
\(\displaystyle \implies \ \ \) \(\displaystyle x^2 + 2 y^2\) \(=\) \(\displaystyle c\)

Hence the result.

$\blacksquare$


Sources