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Pedal equation of an ellipse

Short answer

With pole at centre, the pedal equation is 1/p² = (a² + b² − r²)/(a²b²), equivalently p² = a²b²/(a² + b² − r²).

Find the pedal equation of the ellipse x²/b² + y²/a² = 1.

Assumption: the pole is the centre and a > b; so the semi-axes are b (along x) and a (along y). The result for the focus is also given.

Ellipse: x²/b² + y²/a² = 1. The tangent at (x₁, y₁) is x x₁/b² + y y₁/a² = 1.

The perpendicular from the origin to this line is

p = 1 / √(x₁²/b⁴ + y₁²/a⁴), so 1/p² = x₁²/b⁴ + y₁²/a⁴ …(1)

Also r² = x₁² + y₁² and x₁²/b² + y₁²/a² = 1. Put X = x₁²/b². Then y₁² = a²(1 − X) and

r² = b²X + a²(1 − X), so X = (a² − r²)/(a² − b²).

Now (1) becomes

1/p² = X/b² + (1−X)/a²
     = [a²X + b²(1−X)] / (a²b²)
     = [b² + (a²−b²)X] / (a²b²)
     = (a² + b² − r²) / (a²b²)

Pedal equation (pole at centre): 1/p² = (a² + b² − r²)/(a²b²), i.e. p² = a²b²/(a² + b² − r²).

(With the pole at a focus, the pedal equation is b²/p² = 2a/r − 1.)

✓ Answer checked by Claude Opus 5.5 (high effort)

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