Ellipse

Standard Equation of an Ellipse

• x²/a² + y²/b² = 1, where a > b > 0
• Major axis along x-axis
• Centre: (0, 0)

Symmetry

• Symmetric about x-axis
• Symmetric about y-axis
• Symmetric about origin

Axes & Key Lengths

• Length of major axis: 2a
• Length of minor axis: 2b
• Vertices: (±a, 0)
• Co-vertices: (0, ±b)

Foci & Eccentricity

• Distance of focus from centre: ae
• Foci: (±ae, 0)
• Distance between foci: 2ae
• e = √(1 − b²/a²)
• b² = a²(1 − e²)

Directrices

• Equations: x = ±a/e
• Distance between directrices: 2a/e

Important Property

• For any point P on ellipse: SP + S′P = 2a (constant)

Latus Rectum

• Length of latus rectum: 2b²/a
• Equations: x = ±ae
• Endpoints: (ae, ±b²/a), (−ae, ±b²/a)

Auxiliary Circle

• Equation: x² + y² = a²
• Used to define eccentric angle
• If point on ellipse is (a cosθ, b sinθ), θ is eccentric angle

Parametric Coordinates

• x = a cosθ
• y = b sinθ
• Parameter range: 0 ≤ θ < 2π

Equation of Chord

• Chord joining points θ₁ and θ₂:
• (x/a) cos((θ₁+θ₂)/2) + (y/b) sin((θ₁+θ₂)/2) = cos((θ₁−θ₂)/2)

Position of a Point

• For point (x₁, y₁), evaluate:
• x₁²/a² + y₁²/b²
• > 1 → outside ellipse
• = 1 → on ellipse
• < 1 → inside ellipse

Equation of Tangent

Point Form

• Tangent at (x₁, y₁): xx₁/a² + yy₁/b² = 1

Parametric Form

• Tangent at (a cosθ, b sinθ): (x/a) cosθ + (y/b) sinθ = 1

Slope Form

• y = mx ± √(a²m² + b²)
• Condition of tangency: c² = a²m² + b²

Director Circle

• Locus of intersection of perpendicular tangents
• Equation: x² + y² = a² + b²

Equation of Normal

Point Form

• a²x/x₁ − b²y/y₁ = a² − b²

Parametric Form

• ax/secθ − by/cosecθ = a² − b²

Slope Form

• y = mx ± m(a² − b²)/√(a² + b²m²)

Pair of Tangents from a Point

• From point (x₁, y₁):
• SS₁ = T²
• Where S = x²/a² + y²/b² − 1, S₁ = x₁²/a² + y₁²/b² − 1, T = xx₁/a² + yy₁/b² − 1

Chord with Given Midpoint

• Equation: T = S₁

Chord of Contact

• Equation: T = 0

Pole & Polar

• Polar of point (x₁, y₁): T = 0
• Polar of focus is the directrix
• Conjugate points & lines follow reciprocity property

Diameter of an Ellipse

• Locus of midpoints of parallel chords
• Diameter bisecting chords of slope m:
• y = (b² / a²m) x
• Every diameter passes through centre

Conjugate Diameters

• Each diameter bisects chords parallel to the other
• Major & minor axes are conjugate diameters
• If slopes are m₁, m₂:
• m₁m₂ = −b²/a²
• Eccentric angles differ by π/2