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:

$y^2 = 4 c \paren {x + c}$

Proof

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

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

\(\text {(2)}: \quad\)

\(\ds 2 y \frac {\d y} {\d x}\)

\(=\)

\(\ds 4 c\)

\(\ds \leadsto \ \ \)

\(\ds c\)

\(=\)

\(\ds \frac y 2 \frac {\d y} {\d x}\)

\(\ds \leadsto \ \ \)

\(\ds y^2\)

\(=\)

\(\ds 2 y \frac {\d y} {\d x} \paren {x + \frac y 2 \frac {\d y} {\d x} }\)

substituting for $c$ into $(1)$

\(\ds \)

\(=\)

\(\ds 2 x y \frac {\d y} {\d x} + y^2 \paren {\frac {\d y} {\d x} }^2\)

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

\(\ds y^2\)

\(=\)

\(\ds -2 x y \frac {\d x} {\d y} + y^2 \paren {-\frac {\d x} {\d y} }^2\)

\(\ds \leadsto \ \ \)

\(\ds y^2 \paren {\frac {\d x} {\d y} }^2\)

\(=\)

\(\ds 2 x y \frac {\d x} {\d y} + y^2\)

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Sources

• 1972: George F. Simmons: Differential Equations ... (previous) ... (next): $1$: The Nature of Differential Equations: $\S 3$: Families of Curves. Orthogonal Trajectories: Problem $2$