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An ellipse is the set of all points where the sum of distances to two fixed points (foci) is constant. Standard form: (x-h)2/a2 + (y-k)2/b2 = 1, where a > b means horizontal major axis. The center is (h,k), vertices are a units from the center along the major axis, co-vertices are b units along the minor axis, and foci are c units from the center where c2 = a2 - b2. A circle is a special ellipse where a = b.
Start from the definition with two thumbtacks and a string (physically tracing an ellipse). Derive key relationships: a, b, c, and c2 = a2 - b2. Practice identifying center, vertices, co-vertices, and foci from equations. Graph by plotting these key points. Convert from general form using completing the square. Discuss eccentricity (e = c/a) as a measure of "ovalness."
From circles, you know that a circle is the set of all points at a fixed distance from a single center point. An ellipse is the natural generalization: instead of one fixed point, you use two fixed points called foci (singular: focus), and instead of a fixed distance to one point, you require that the *sum* of distances to both foci is constant. Imagine pinning two thumbtacks in a corkboard, looping a string around them, pulling the string taut with a pencil, and tracing a curve — the resulting shape is an ellipse. As the two foci get closer together, the ellipse approaches a circle; when they coincide, it *is* a circle. This is why the circle is a special case of the ellipse (a = b, so c = 0, meaning both foci are at the center).
The standard equation (x−h)²/a² + (y−k)²/b² = 1 encodes the geometry. The center is (h, k). The parameter a is always the larger denominator — it gives the distance from the center to the vertices, the endpoints of the longer (major) axis. The parameter b gives the distance to the co-vertices, the endpoints of the shorter (minor) axis. The foci are located at distance c from the center along the major axis, where c² = a² − b². This relationship follows from the Pythagorean theorem applied to a right triangle formed by a, b, and c at a special point on the ellipse: at an endpoint of the minor axis, the two distances to the foci each equal a (since their sum equals 2a and by symmetry they are equal), so a² = b² + c².
To identify the orientation, look at which denominator is larger. If a² is under (x−h)², the major axis runs horizontally and the vertices are to the left and right of center. If a² is under (y−k)², the major axis is vertical. A common mistake is to assume x always gets a² — it does not. The rule is simply: a is the bigger number, whichever variable it goes with, and the major axis runs in that direction.
The eccentricity e = c/a measures how "stretched" the ellipse is, ranging from 0 (a perfect circle, c = 0) to just below 1 (a very elongated ellipse, c approaching a). Earth's orbit is an ellipse with eccentricity about 0.017 — nearly circular, with the Sun at one focus. Halley's comet has eccentricity about 0.97 — a dramatically elongated ellipse. Understanding eccentricity connects the geometric definition (ratio of distances) to the algebraic parameters and prepares you for hyperbolas, where eccentricity exceeds 1.