What Are Conic Sections?
A conic section is any curve you get by slicing a double cone with a plane. Depending on the angle of the slice, you get one of four shapes: a circle, an ellipse, a parabola, or a hyperbola.
These four curves were studied systematically by the ancient Greeks (particularly Apollonius around 200 BCE) long before anyone had algebraic equations for them. Today we recognize conic sections as the curves satisfying second-degree equations in two variables: Ax² + Bxy + Cy² + Dx + Ey + F = 0. By choosing different coefficients, you get each of the four types.
e classifies conics more cleanly: e = 0 gives a circle, 0 < e < 1 gives an ellipse, e = 1 gives a parabola, and e > 1 gives a hyperbola.Where conics show up
Conic sections appear throughout science and engineering: planetary orbits are ellipses, projectile paths are parabolas (ignoring air resistance), the cross-section of a satellite dish is parabolic, and the paths of some unbound comets are hyperbolas. Lens design, antenna shapes, architecture, and acoustics all depend on properties of these curves.
Circles
A circle is the set of all points equidistant from a fixed center.
(h, k) and radius r is (x − h)² + (y − k)² = r². If the center is at the origin, this simplifies to x² + y² = r².Graph (x − 2)² + (y + 3)² = 16.
The center is at (2, −3) (note the sign flip: (y + 3) is (y − (−3))).
The radius is √16 = 4.
Draw a circle of radius 4 around (2, −3).
Center (2, −3), radius 4Converting general form to standard form
Given x² + y² + Dx + Ey + F = 0, complete the square in x and in y to rewrite as (x − h)² + (y − k)² = r².
Rewrite x² + y² − 6x + 4y − 12 = 0.
Group: (x² − 6x) + (y² + 4y) = 12.
Complete the square: (x² − 6x + 9) + (y² + 4y + 4) = 12 + 9 + 4 = 25.
Standard form: (x − 3)² + (y + 2)² = 25.
Center (3, −2), radius 5Circles are the simplest conic — eccentricity 0, fully symmetric, with one parameter (radius) determining the size.
Ellipses
An ellipse is the set of all points where the sum of distances to two fixed points (the foci) is constant.
(h, k) is (x − h)²/a² + (y − k)²/b² = 1, where a > b > 0. a is the semi-major axis (half the long diameter), b is the semi-minor axis. The foci lie on the major axis, at distance c from the center, where c² = a² − b².Analyze x²/25 + y²/9 = 1.
Here a² = 25 (so a = 5), b² = 9 (so b = 3).
The major axis is horizontal (because a² > b² is under x²).
c² = 25 − 9 = 16, so c = 4.
Foci: (±4, 0). Vertices: (±5, 0). Co-vertices: (0, ±3).
e = c/a = 4/5 = 0.8(x − h)²/b² + (y − k)²/a² = 1, with a > b. Major axis is vertical; foci are at (h, k ± c).Kepler's First Law says planets orbit the Sun in ellipses with the Sun at one focus. The eccentricity of Earth's orbit is about 0.0167 — very close to circular. Halley's Comet has eccentricity ≈ 0.967, so its orbit is a highly elongated ellipse.
Parabolas
A parabola is the set of all points equidistant from a fixed point (the focus) and a fixed line (the directrix).
(h, k) is (x − h)² = 4p(y − k). If p > 0, it opens upward; if p < 0, downward. The focus is at (h, k + p), and the directrix is the horizontal line y = k − p. The magnitude |p| is the distance from the vertex to the focus (and from the vertex to the directrix).(y − k)² = 4p(x − h). Opens rightward if p > 0, leftward if p < 0. Focus (h + p, k), directrix x = h − p.Analyze x² = 8y.
This is vertex-at-origin form with 4p = 8, so p = 2.
The parabola opens upward.
Focus: (0, 2), Directrix: y = −2Analyze (y − 3)² = −12(x − 1).
Vertex: (1, 3).
4p = −12, so p = −3. Horizontal, opens leftward.
Focus: (1 + (−3), 3) = (−2, 3). Directrix: x = 1 − (−3) = 4.
Focus: (−2, 3), Directrix: x = 4Projectile motion (ignoring air resistance), satellite dish reflectors (parallel rays reflect to the focus), and the cables of suspension bridges. Any time you see a perfectly symmetric arc, you're likely looking at a parabola.
Hyperbolas
A hyperbola is the set of all points where the difference of distances to two fixed foci is constant (in absolute value). A hyperbola has two disconnected branches.
(h, k) is (x − h)²/a² − (y − k)²/b² = 1. Key features: The vertices are at (h ± a, k). The foci are at (h ± c, k), where c² = a² + b² (note the PLUS sign — different from an ellipse). The asymptotes are lines y − k = ±(b/a)(x − h). The hyperbola approaches but never touches these lines.(y − k)²/a² − (x − h)²/b² = 1. Vertices at (h, k ± a), foci at (h, k ± c), asymptotes y − k = ±(a/b)(x − h).Analyze x²/16 − y²/9 = 1.
a² = 16 (so a = 4), b² = 9 (so b = 3). Horizontal hyperbola centered at origin.
Vertices: (±4, 0).
c² = 16 + 9 = 25, c = 5. Foci: (±5, 0).
Asymptotes: y = ±(3/4)x.
e = c/a = 5/4 = 1.25The path of an object escaping a gravitational field, hyperbolic navigation (LORAN), cooling tower cross-sections, and certain optical instruments.
The identifying algebraic feature is the minus sign between the x² and y² terms — that's what separates hyperbolas from ellipses.
Identifying Conic Sections from General Form
Given a general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, you can identify which conic it represents by examining the coefficients.
For equations without an xy term (B = 0), the rules are simple:
| Conic | Condition on A and C |
|---|---|
| Circle | A = C (and both nonzero) |
| Ellipse | A and C have the same sign but are not equal |
| Parabola | Exactly one of A or C is zero |
| Hyperbola | A and C have opposite signs |
Identify 4x² + 9y² − 8x + 36y + 4 = 0.
A = 4, C = 9, same sign, not equal.
EllipseIdentify x² − 4y² + 6x + 24y − 43 = 0.
A = 1, C = −4, opposite signs.
HyperbolaIdentify y² + 4x − 6y + 13 = 0.
A = 0 (no x²), C = 1 (nonzero).
ParabolaWhen there's an xy term
B² − 4AC. If B² − 4AC < 0, it's an ellipse (or circle). If B² − 4AC = 0, it's a parabola. If B² − 4AC > 0, it's a hyperbola.Most pre-calculus problems avoid the xy term, so the coefficient rule is usually enough.
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