Three conics — From loci to equations

Mode

Drag P and watch the sum of the two distances.

a — half the vertex separation5

c — distance OF3

Movement of P

Position of P(0, 4)

A constant distance sum

PF₁ + PF₂

≈ 10 units

Condition on the locus: 2a = 10 units

P ≈ (0, 4)

P is constrained to the locus

Substitute P:
Left side ≈ 1; Right side ≈ 1

From distances to an equation

PF₁ and PF₂ change, but their sum stays equal to 2a. Both foci lie inside the ellipse.

For a > c > 0, let b² = a² − c².

√((x + c)² + y²) + √((x − c)² + y²) = 2a

Distance between foci F₁F₂ = 2c: 6 units

Model notes

This resource uses standard equations with a horizontal principal axis and the center or vertex at the origin.
Hyperbolas and parabolas are unbounded; the drawing and trace show only the portion under study.
Tracing speed is a presentation choice, not physical motion. Displayed measurements are rounded.
For a freely moved P, “approximately satisfies” means a distance residual no greater than 0.02 units; it does not establish an exact solution.

Mathematics

Three conics — From loci to equations

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Resource information and sources

Shape and foci