Coordinate Geometry
Ellipse
Types of ellipses, eccentricity, tangents, normals, and focal distances.
An ellipse is the locus of points whose distances to two fixed foci sum to a constant. The two foci lie on the major axis, equidistant from the centre; the constant sum equals the major-axis length 2a. The semi-major axis a and semi-minor axis b set the size and shape; the eccentricity e ∈ [0, 1) measures elongation: e = 0 is a circle, e → 1 a near-degenerate slit. The two standard orientations place the centre at the origin and align the major axis with either coordinate axis.
Types of Ellipse
The two standard orientations swap which axis carries the longer semi-axis; the behaviour is symmetric under that swap.
Type 1 has the major axis along x, so a > b. The vertices on the major axis are (±a, 0); the co-vertices on the minor axis are (0, ±b). The foci lie on the x-axis at (±ae, 0). The curve is symmetric in both axes and in the origin.
Type 2 has the major axis along y, so b > a. The denominators keep their positions, but b is now the semi-major axis and a the semi-minor. The foci move onto the y-axis at (0, ±be); the vertices on the major axis become (0, ±b).
| Ellipse | Type 1 (a>b) | Type 2 (b>a) |
|---|---|---|
| Center | (0,0) | (0,0) |
| Vertices | A(a,0) A’(-a,0) B(0,b) B’(0,-b) | B(0,b) B’(0,-b) A(a,0) A’(-a,0) |
| Foci | S=(ae,0) S”(-ae,0) | S=(be,0) S”(-be,0) |
| Distance Between Foci | 2ea | 2be |
| Length of Axis | Major = 2a, Minor = 2b | Major = 2b, Minor = 2a |
| Equation of Axis | Major: y=0, Minor: x=0 | Major: x=0, Minor: y=0 |
| Relation Between a,b,c | b2 = a2(1-e2) | a2 = b2(1-e2) |
Eccentericity (Type 1)
For Type 1 the eccentricity comes from the two semi-axes: e = √(a2 − b2) / a. Since a > b the radicand is positive and e lies strictly between 0 and 1. The quantity ae is the centre-to-focus distance, so e measures focal offset as a fraction of the semi-major axis.
Eccentericity (Type 2)
For Type 2 the major axis is b, so the eccentricity formula swaps the roles of a and b under the radical. The radicand b2 − a2 is positive because b > a. The result lies in [0, 1), with the lower bound (a circle) reached when a = b.
Tangent to Ellipse
Tangent lines come in point-based and slope-based forms, as for the circle and parabola. The point form applies T = 0 to the ellipse equation: x2 becomes x x1 and y2 becomes y y1. The contact point (x1, y1) must satisfy the original ellipse equation, otherwise the resulting line is a polar rather than a tangent.
The slope form gives the two parallel tangents of slope m. Substituting y = mx + c into the ellipse equation and demanding a double root yields c2 = a2 m2 + b2. The radicand is always positive, so a tangent of every slope exists (unlike the hyperbola).
Normal to Ellipse
The normal at (x1, y1) is perpendicular to the tangent and passes through the contact point. Unlike the circle, the ellipse normal does not generally pass through the centre; it does so at the four vertices, and elsewhere reflects rays from one focus to the other. The form below is undefined when x1 or y1 is zero (at a vertex or co-vertex), where the normal is a coordinate axis.
Focal Distance
Each point on the ellipse has two focal distances, one to each focus, summing to 2a (the defining locus property). For Type 1 the distance to the focus at (ae, 0) is |a − e x1| and the distance to (−ae, 0) is |a + e x1|. Both reduce to a at the centre (x1 = 0). The form below assumes Type 1; swap x1 for y1 and a for b in Type 2.