# Orthogonal Trajectories/Parabolas Tangent to X Axis

## 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:

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

## Proof

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

$x \dfrac {\mathrm d y} {\mathrm d x} + y = 0$
 $\text {(2)}: \quad$ $\displaystyle \frac {\mathrm d y} {\mathrm d x}$ $=$ $\displaystyle 2 c x$ $\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.

$\blacksquare$