Orthogonal Trajectories/Examples/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:
 * $x^2 + 2 y^2 = c$


 * ParabolasTangentAxisOrthogonalTrajectories.png

Proof
We use the technique of formation of ordinary differential equation by elimination.

Differentiating $(1)$ $x$ gives:
 * $x \dfrac {\d y} {\d x} + y = 0$

Thus from Orthogonal Trajectories of One-Parameter Family of Curves, the family of orthogonal trajectories is given by:
 * $\dfrac {\d y} {\d x} = -\dfrac x {2 y}$

So:

Hence the result.