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Analytic Geometry2D

The Ellipse - Theory & Concepts - Analytic Geometry Ellipse Foci Construction

Equations of ellipses, finding foci, vertices, eccentricity, and area.

Open the complete lesson

Ellipse: Foci String Construction Explorer

Geometric parameters

Semi-major axis a (string half-length)6
Focal distance c (focus location)4

Tracing Angle

Angle θ (theta)45°

Construction verification

Ellipse Formula
x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
a=6.0, b=a2−c2=62−42=4.47a = 6.0, \ b = \sqrt{a^2 - c^2} = \sqrt{6^2 - 4^2} = 4.47
Foci: F1(−4,0), F2(4,0)F_1(-4, 0), \ F_2(4, 0)
Distances to Foci
d1=PF1=8.828, d2=PF2=3.172d_1 = PF_1 = 8.828, \ d_2 = PF_2 = 3.172
Pins-and-String Sum
d1+d2=8.828+3.172=12.0=2ad_1 + d_2 = 8.828 + 3.172 = 12.0 = 2a
The total string length remains exactly 12 units at all angles.
Focus F1 (-4, 0)F₁Focus F2 (4, 0)F₂P (4.2, 3.2)P(4.2, 3.2)

Move the "Angle θ" slider to trace the ellipse. The indigo string adjusts dynamically.

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