DWG NO. 8.1 — Lesson 1 of 4
Unit 8: Conic Sections · ~25 min
A circle is the simplest conic: every point the same distance from a center. An ellipse relaxes that rule slightly — instead of one fixed point, it uses two, called foci (singular: focus), and instead of a fixed distance, it fixes the sum of the two distances to those foci. Squash the two foci together into a single point and the sum-of-distances rule collapses back into the circle's single-distance rule — the circle is really just an ellipse whose foci coincide.
A circle is the set of all points at a fixed distance r (the radius) from a center (h, k).
A circle's general form, x² + y² + Dx + Ey + F = 0, hides the center and radius inside the coefficients. Completing the square on both x and y recovers standard form — the same move you used on quadratics back in Unit 2, just applied twice.
An ellipse is the set of all points P for which the sum of the distances from P to two fixed foci is constant. That constant sum turns out to equal 2a, where a is the distance from the center to a vertex along the ellipse's longer axis (the major axis). The shorter axis is the minor axis, with half-length b.
| Orientation | Standard form | Foci & vertices lie on |
|---|---|---|
| Major axis horizontal | (x−h)\(\frac{^{2}}{a^{2}}\) + (y−k)\(\frac{^{2}}{b^{2}}\) = 1 | the horizontal line y = k |
| Major axis vertical | (x−h)\(\frac{^{2}}{b^{2}}\) + (y−k)\(\frac{^{2}}{a^{2}}\) = 1 | the vertical line x = h |
In both cases a > b, and the larger denominator tells you which axis is major. The distance from center to focus, c, satisfies c² = a² − b² — the foci always sit strictly inside the vertices, closer to the center.
Ellipse \(\frac{x^{2}}{25}\) + \(\frac{y^{2}}{9}\) = 1 — a = 5, b = 3, c = \(\sqrt{25-9}\) = 4, foci at (±4, 0)