Orthogonal Trajectories/Examples/Parabolas with Focus at Origin

Theorem
Consider the one-parameter family of curves of parabolas whose focus is at the origin and whose axis is the $x$-axis:
 * $(1): \quad y^2 = 4 c \paren {x + c}$

Its family of orthogonal trajectories is given by the equation:
 * $y^2 = 4 c \paren {x + c}$


 * ParabolasFocusOriginOrthogonalTrajectories.png

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

Differentiating $(1)$ $x$ gives:

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