Conic Sections
Easy Overview
Take a double cone — two ice-cream cones meeting at their tips — and slice it with a plane. If you cut straight across, you get a circle. Tilt the plane a bit and you get an ellipse. Tilt it so it's parallel to the slant of the cone and you get a parabola. Tilt it steeper so it cuts both halves of the cone and you get a hyperbola — two separate mirrored curves. These are the conic sections, and they're some of the most beautiful curves in mathematics. The ancient Greeks discovered conic sections around 200 BCE. They were studied as pure geometry for centuries before anyone found practical uses. Today, they're everywhere. Parabolas describe the path of a thrown ball, the shape of satellite dishes, and the reflectors in car headlights. Ellipses describe planetary orbits (Kepler's first law) and the shape of whispering galleries. Hyperbolas appear in navigation (LORAN systems), optics, and the trajectories of particles in nuclear physics. Each conic section is defined by its eccentricity e — a number that describes how 'stretched' the curve is. A circle has e = 0. An ellipse has 0 < e < 1. A parabola has e = 1 exactly. A hyperbola has e > 1. The eccentricity determines the shape completely. We'll start with the parabola: the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). Its standard form y² = 4ax (opening right) gives focus at (a, 0) and directrix x = −a. The vertex is at (0, 0). The latus rectum — a chord through the focus perpendicular to the axis — has length 4a. Next, the ellipse: the set of points where the sum of distances to two foci is constant. Its standard form x²/a² + y²/b² = 1 (a > b) has major axis along the x-axis of length 2a, minor axis of length 2b, and foci at (±ae, 0) where e = √(1 − b²/a²). Finally, the hyperbola: the set of points where the absolute difference of distances to two foci is constant. Its standard form x²/a² − y²/b² = 1 has foci at (±ae, 0) with e = √(1 + b²/a²) > 1. Hyperbolas have asymptotes — diagonal lines the curve approaches: y = ±(b/a)x. By the end, you'll be able to identify any conic from its equation, find its key features (focus, directrix, vertices, axes, eccentricity, latus rectum), and understand the geometric relationships that define each curve.
What are conic sections — slicing the cone
A double cone is formed by rotating a line (generator) about an axis. A plane slicing this cone creates different curves. The type of curve depends on the angle of the cut: horizontal → circle, slightly tilted → ellipse, parallel to the generator → parabola, steep enough to cut both cones → hyperbola. The intersection of a plane and a cone is called a 'conic section'. If the plane passes through the vertex, the conic is degenerate (a point, line, or pair of lines).
Eccentricity — the shape descriptor
Eccentricity e measures how much a conic deviates from being circular. For any conic: e = distance from any point to focus / distance from that point to directrix. Circle: e = 0. Ellipse: 0 < e < 1. Parabola: e = 1. Hyperbola: e > 1. The eccentricity completely determines the shape — two conics with the same e are similar (same shape, different sizes).
The parabola — equidistant from focus and directrix
A parabola is the set of all points P such that distance from P to focus F equals distance from P to directrix ℓ. Standard form opening right: y² = 4ax. Focus: (a, 0). Directrix: x = −a. Vertex: (0, 0). Axis: x-axis. Latus rectum: chord through focus perpendicular to axis, length 4a. If the x and y are swapped (x² = 4ay), the parabola opens upward. If coefficients are negative, it opens left or downward.
Standard forms of the parabola
There are four standard orientations: y² = 4ax (opens right, focus (a,0), directrix x = −a). y² = −4ax (opens left, focus (−a,0), directrix x = a). x² = 4ay (opens up, focus (0,a), directrix y = −a). x² = −4ay (opens down, focus (0,−a), directrix y = a). In each case, the vertex is at (0,0) and the latus rectum length is |4a|. The axis is the line through the focus and vertex.
Tangent to a parabola
For the parabola y² = 4ax, the tangent at point (x₁, y₁) is yy₁ = 2a(x + x₁). The slope form: y = mx + a/m (where m ≠ 0) is a tangent, and it touches at (a/m², 2a/m). The condition for a line y = mx + c to be tangent to y² = 4ax is c = a/m. For the parametric point (at², 2at), the tangent is ty = x + at².
The ellipse — constant sum of distances
An ellipse is the set of points P such that PF₁ + PF₂ = 2a (constant), where F₁ and F₂ are the two foci. Standard form (horizontal major axis): x²/a² + y²/b² = 1, where a > b. Centre: (0,0). Vertices: (±a, 0). Co-vertices: (0, ±b). Foci: (±c, 0) where c² = a² − b². Eccentricity e = c/a = √(1 − b²/a²). Length of major axis: 2a. Length of minor axis: 2b. Latus rectum length: 2b²/a.
Key properties of the ellipse
The ellipse x²/a² + y²/b² = 1 (a > b) has several notable features. The sum of distances from any point on the ellipse to the two foci is constant (= 2a). This is the defining property. If a < b, the major axis is vertical, and the ellipse is x²/a² + y²/b² = 1 with foci at (0, ±c) where c² = b² − a². The eccentricity still e = c/a (using the larger denominator under the square root). The area of the ellipse is πab.
Tangent to an ellipse
For the ellipse x²/a² + y²/b² = 1, the tangent at (x₁, y₁) is xx₁/a² + yy₁/b² = 1. The slope form: y = mx ± √(a²m² + b²) is a tangent. Condition: c² = a²m² + b² for the line y = mx + c to be tangent. The parametric form (a cos θ, b sin θ) gives the tangent: (x cos θ)/a + (y sin θ)/b = 1.
The hyperbola — constant difference of distances
A hyperbola is the set of points P such that |PF₁ − PF₂| = 2a (constant). Standard form (horizontal transverse axis): x²/a² − y²/b² = 1. Centre: (0,0). Vertices: (±a, 0). Foci: (±c, 0) where c² = a² + b². Eccentricity e = c/a = √(1 + b²/a²) > 1. Latus rectum length: 2b²/a. The hyperbola has two branches — one on each side of the y-axis.
Asymptotes of a hyperbola
A unique feature of hyperbolas: they have asymptotes — straight lines the curve approaches as x → ±∞. For x²/a² − y²/b² = 1, the asymptotes are y = ±(b/a)x. The hyperbola gets arbitrarily close to these lines but never touches them. The asymptotes pass through the centre and form a 'X' shape. The hyperbola lies entirely within the two opposite angles formed by the asymptotes. If a = b, the asymptotes are y = ±x (perpendicular), and the hyperbola is called rectangular.
Conjugate hyperbola
The conjugate hyperbola of x²/a² − y²/b² = 1 is x²/a² − y²/b² = −1, or equivalently y²/b² − x²/a² = 1. The conjugate shares the same asymptotes but its transverse axis is vertical instead of horizontal. Its vertices are at (0, ±b) instead of (±a, 0). The pair (hyperbola + conjugate) together with their asymptotes form a complete geometric picture centred at the origin.
Rectangular hyperbola
A rectangular hyperbola is one where the asymptotes are perpendicular (at right angles). This happens when a = b, so the equation is x² − y² = a². The asymptotes are y = ±x, which are perpendicular. The eccentricity of a rectangular hyperbola is e = √2. Another common form is xy = c², which is a rectangular hyperbola rotated by 45°. Its asymptotes are the coordinate axes themselves.
Tangent to a hyperbola
For the hyperbola x²/a² − y²/b² = 1, the tangent at (x₁, y₁) is xx₁/a² − yy₁/b² = 1. The slope form: y = mx ± √(a²m² − b²). Condition for y = mx + c to be tangent: c² = a²m² − b² (note the minus, different from the ellipse). The parametric form (a sec θ, b tan θ) gives the tangent: (x sec θ)/a − (y tan θ)/b = 1.
Identifying conics from their equations
A second-degree equation ax² + by² + 2hxy + 2gx + 2fy + c = 0 represents a conic. The type is determined by the discriminant Δ = h² − ab (or just look at the signs). If h = 0 and a = b: circle. If Δ < 0 (and h = 0, a ≠ b): ellipse. If Δ = 0: parabola. If Δ > 0: hyperbola. If a + b = 0 and h = 0: rectangular hyperbola. The presence of an xy term means the conic is rotated.
Applications of conic sections in real life
Parabolas: satellite dishes and telescopes (parallel rays converge at focus), car headlights (light from focus reflects as parallel beam), projectile motion (balls, rockets, water fountains). Ellipses: planetary orbits (Kepler's first law), elliptical pool tables (ball from one focus bounces to the other), whispering galleries (sound from one focus is heard at the other). Hyperbolas: navigation (LORAN time difference), cooling towers of nuclear plants (hyperboloid shape for strength), sonic booms (Mach cone forms a hyperbola on the ground).
Parametric forms of conics
Parametric equations are often more convenient than Cartesian. Parabola y² = 4ax: (at², 2at). Ellipse x²/a² + y²/b² = 1: (a cos θ, b sin θ). Hyperbola x²/a² − y²/b² = 1: (a sec θ, b tan θ). The parameter t or θ varies over the domain. This form is especially useful for deriving tangents, normals, and other properties.
Latus rectum of conics
The latus rectum is a chord through the focus perpendicular to the axis. For parabola y² = 4ax: length = 4a. For ellipse x²/a² + y²/b² = 1: length = 2b²/a. For hyperbola x²/a² − y²/b² = 1: length = 2b²/a. The endpoints of the latus rectum are important points on the conic and are often used in problem-solving.
Key Points
- •All conics come from slicing a double cone with a plane
- •Eccentricity e: circle (0), ellipse (0<e<1), parabola (1), hyperbola (e>1)
- •Parabola: set of points equidistant from focus and directrix
- •Standard parabola y² = 4ax: focus (a,0), directrix x = −a, vertex (0,0)
- •Latus rectum of parabola = 4a
- •Ellipse: sum of distances to foci = 2a (constant)
- •Standard ellipse x²/a² + y²/b² = 1: foci (±c,0), c² = a² − b², e = c/a
- •Hyperbola: |difference of distances to foci| = 2a (constant)
- •Standard hyperbola x²/a² − y²/b² = 1: foci (±c,0), c² = a² + b², e = c/a > 1
- •Asymptotes of hyperbola x²/a² − y²/b² = 1: y = ±(b/a)x
- •Rectangular hyperbola: a = b (asymptotes at 90°), or xy = c²
- •Tangent to parabola y² = 4ax at (x₁, y₁): yy₁ = 2a(x + x₁)
- •Tangent to ellipse at (x₁, y₁): xx₁/a² + yy₁/b² = 1
- •Tangent to hyperbola at (x₁, y₁): xx₁/a² − yy₁/b² = 1
- •General conic discriminant: h² − ab < 0 (ellipse), = 0 (parabola), > 0 (hyperbola)
Practice Questions
- Find the focus, directrix, vertex, and length of latus rectum of the parabola y² = 16x.
- Find the equation of an ellipse with foci at (±4, 0) and major axis of length 10.
- Find the eccentricity, foci, and asymptotes of the hyperbola x²/9 − y²/16 = 1.
- A bridge is in the shape of a parabolic arch. The span is 30 m and the maximum height is 10 m. Find the equation of the parabola.
- Prove that for any point on an ellipse, the sum of distances to the foci is constant.
- Find the equation of the hyperbola with vertices at (±3, 0) and foci at (±5, 0).
- Find the equation of the tangent to the ellipse x²/16 + y²/9 = 1 at the point (4 cos θ, 3 sin θ).
- Identify the conic: x² + 4y² − 6x + 8y − 3 = 0. Find its centre and eccentricity.