Conic sections
The conic sections are the curves obtained by cutting a cone with a plane: the circle, ellipse, parabola and hyperbola. Each is the locus of a point whose distance from a fixed point (the focus) is a constant multiple (the eccentricity e) of its distance from a fixed line (the directrix).
| Conic | Eccentricity | Standard equation | Key features |
|---|---|---|---|
| Parabola | e = 1 | y² = 4ax | Focus (a, 0), directrix x = −a |
| Ellipse | e < 1 | x²/a² + y²/b² = 1 (a > b) | Foci (±ae, 0), e² = 1 − b²/a² |
| Hyperbola | e > 1 | x²/a² − y²/b² = 1 | Foci (±ae, 0), asymptotes y = ±bx/a, e² = 1 + b²/a² |
| Circle | e = 0 | x² + y² = r² | Centre at the origin |
The parabola
The ellipse
An ellipse is the locus of a point P for which PF₁ + PF₂ = 2a, a constant. The sum of the distances from the two foci is the same everywhere. Its area is πab.
Choose the semi-axes a ≥ b.
The hyperbola
A hyperbola is the locus of P for which |PF₁ − PF₂| = 2a. It has two branches and two asymptotes y = ±(b/a)x. The rectangular hyperbola xy = c² has the axes as asymptotes.
Polar coordinates
A point can be located by its distance r from the origin and the angle θ from the positive x-axis: (r, θ). To convert, x = r cos θ, y = r sin θ, r² = x² + y², tan θ = y/x. For example the circle r = 4 and the line θ = π/4. The curve r = a(1 + cos θ) is a cardioid.
Key terms
- Focus
- The fixed point of a conic.
- Directrix
- The fixed line of a conic.
- Eccentricity
- The ratio of the distance from the focus to the distance from the directrix.
- Polar coordinates
- Position given by distance and angle.
Summary
- Conics are circles, ellipses, parabolas and hyperbolas, distinguished by eccentricity.
- y² = 4ax; x²/a² ± y²/b² = 1.
- An ellipse has constant PF₁ + PF₂; a hyperbola has constant |PF₁ − PF₂|.
- Polar and Cartesian coordinates are linked by x = r cos θ, y = r sin θ.