Ellipse

Major axis along the x-axis, length 2a. Minor axis along the y-axis, length 2b. Vertices at (±a, 0) and (0, ±b). Foci at (±c, 0) where c² = a² − b².

Major axis along the y-axis, length 2a. Minor axis along the x-axis, length 2b. Vertices at (0, ±a) and (±b, 0). Foci at (0, ±c) where c² = a² − b².

Centre at (h, k). All elements of the standard ellipse apply with origin shifted to (h, k). Identified by completing the square in the general second-degree equation.

Where c is the focal distance (distance from centre to focus) and e is the eccentricity. Both forms are equivalent; the second is used when switching between geometric and eccentricity-based descriptions.

Eccentricity measures the deviation from a circle. e → 0 gives a circle (a = b); e → 1 gives a parabola in the limit (b → 0). Directrices are at x = ±a/e.

Distances from P(x₁, y₁) on the ellipse to the two foci S(−c, 0) and S′(c, 0). r₁ is the distance to the nearer focus when x₁ > 0. Both are positive since −a ≤ x₁ ≤ a.

The defining property of an ellipse: the sum of distances from any point on the ellipse to the two foci is constant, equal to the major axis length 2a.

Each focus has a latus rectum passing through it perpendicular to the major axis. Endpoints of the latus rectum through (c, 0) are (c, ±b²/a). The semi-latus rectum b²/a is the harmonic mean of the two focal segments of any focal chord.

θ is the eccentric angle — not the actual angle subtended at the centre. Every point on the ellipse corresponds to a unique θ. The parametric form simplifies tangent, normal, and focal chord calculations.

The circle with the major axis as diameter. If Q(a cosθ, a sinθ) is on the auxiliary circle, the corresponding point P(a cosθ, b sinθ) on the ellipse has the same eccentric angle θ. The ellipse is a uniform vertical compression of the auxiliary circle by factor b/a.

Tangent to x²/a² + y²/b² = 1 at point (x₁, y₁) lying on it. Obtained by the T = 0 rule: replace x² → xx₁, y² → yy₁.

Tangent to the ellipse at the parametric point (a cosθ, b sinθ). Slope of this tangent is −(b cosθ)/(a sinθ) = −b cotθ/a.

Two tangents to x²/a² + y²/b² = 1 exist for each slope m. Point of contact: (∓a²m/√(a²m²+b²), ±b²/√(a²m²+b²)).

Line y = mx + c is tangent to x²/a² + y²/b² = 1 iff c² = a²m² + b². Unlike a parabola, two values of c (opposite signs) work for each slope m.

Normal to x²/a² + y²/b² = 1 at (x₁, y₁). Slope of normal = a²y₁/(b²x₁). The normal passes through neither focus in general.

Normal at (a cosθ, b sinθ). Slope of normal = (a sinθ)/(b cosθ) = (a/b) tanθ. Passes through the centre only when θ = 0°, 90°, 180°, 270° (the vertices).

Normal to x²/a² + y²/b² = 1 with slope m. Up to four normals can be drawn from a point inside the evolute of the ellipse.

Chord joining the two points of tangency when tangents are drawn from external point (x₁, y₁). Same form as the tangent at a point — context determines interpretation.

Combined equation of tangents from (x₁, y₁) to x²/a² + y²/b² = 1, where S = x²/a² + y²/b² − 1, S₁ = x₁²/a² + y₁²/b² − 1, T = xx₁/a² + yy₁/b² − 1.

Chord of the ellipse whose midpoint is (x₁, y₁). Explicitly: xx₁/a² + yy₁/b² − 1 = x₁²/a² + y₁²/b² − 1, i.e. T = S₁.

Locus of points from which the two tangents to the ellipse are perpendicular. A concentric circle of radius √(a²+b²). For a circle (a = b), the director circle has radius a√2.

Polar of (x₁, y₁) with respect to the ellipse. Coincides with the chord of contact when the point is external, and with the tangent when the point is on the ellipse. La Hire's theorem holds.

For any focal chord, the semi-latus rectum b²/a is the harmonic mean of the two focal distances r₁ and r₂ of its endpoints. Equivalently, 1/r₁ + 1/r₂ = 2a/b².

Two diameters y = m₁x and y = m₂x of the ellipse x²/a² + y²/b² = 1 are conjugate iff m₁m₂ = −b²/a². Each bisects chords parallel to the other. For a circle, conjugate diameters are perpendicular (b=a gives m₁m₂=−1).

The ellipse x²/a² + y²/b² = 1 encloses area πab. Reduces to πa² for a circle (b = a). Derived by scaling the unit circle uniformly in one direction.