Conic Sections

Summary

Conic sections—circles, ellipses, parabolas, and hyperbolas—arise as plane slices of a cone and as quadratic curves in the plane. Standard forms make centers, foci, directrices, and axes readable after translation (often via completing the square).

Prerequisites

Distance and Midpoint, Completing the Square. Helpful: Polynomials and Rational Functions. Hub: Analytic Geometry.

Object / Concept

A conic section is the set of points satisfying a second-degree equation

A x^{2} + B x y + C y^{2} + D x + E y + F = 0

(with not all of $A, B, C$ zero). When $B = 0$ and axes are aligned with the coordinate axes, the standard forms below apply after translation.

Coordinate System

Cartesian plane; standard forms assume axes parallel to the coordinate axes (no $x y$ term).

Notation

Symbol	Meaning
$(h, k)$	center or vertex (as appropriate)
$a, b$	semi-axis lengths (ellipse/hyperbola)
$p$ or $4 p$	focal length parameter for parabolas
$r$	radius of a circle

Conditions / Assumptions

Real points: some coefficient choices give empty sets, points, or degenerate pairs of lines.
Forms below assume no rotation ( $B = 0$ ).
For ellipse/hyperbola, $a > 0$ , $b > 0$ ; for circle, $r > 0$ .

Equations

Circle (center $(h, k)$ , radius $r > 0$ )

(x - h)^{2} + (y - k)^{2} = r^{2} .

Ellipse (center $(h, k)$ , axes parallel to coordinate axes)

\frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1.

If $a > b$ , major axis is horizontal; if $b > a$ , major axis is vertical. For a circle, $a = b = r$ .

Parabola (vertex $(h, k)$ )

\begin{aligned} (y - k)^{2} & = 4 p (x - h) & (opens horizontally), \\ (x - h)^{2} & = 4 p (y - k) & (opens vertically) . \end{aligned}

Focus is a distance $| p |$ from the vertex along the axis of symmetry.

Hyperbola (center $(h, k)$ )

\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1 (opens horizontally),

\frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1 (opens vertically) .

Asymptotes (horizontal-opening case):

y - k = \pm \frac{b}{a} (x - h) .

Visual / Geometric Interpretation

Circle: constant distance from a center.
Ellipse: sum of distances to two foci is constant.
Parabola: equidistant from focus and directrix.
Hyperbola: absolute difference of distances to two foci is constant.

Worked Example

Identify $\frac{(x - 1)^{2}}{9} + \frac{(y + 2)^{2}}{4} = 1$ .

Ellipse, center $(1, - 2)$ .
$a^{2} = 9$ , $b^{2} = 4$ so $a = 3$ , $b = 2$ .
Major axis horizontal; vertices at $(1 \pm 3, - 2)$ , i.e. $(4, - 2)$ and $(- 2, - 2)$ .

Common Mistakes

Swapping $a^{2}$ and $b^{2}$ roles when deciding major-axis orientation for ellipses.
Treating $x^{2} / a^{2} - y^{2} / b^{2} = - 1$ without rewriting into a standard hyperbola form.
Forgetting to complete the square before reading $(h, k)$ .
Using $r^{2}$ as radius instead of $r$ .

Connections

Related: Completing the Square, Distance and Midpoint, Lines and Planes
Next: quadratic forms and eigenvalues in Eigenvalues and Eigenvectors
Calculus: conics as level curves and optimization constraints

References

Standard forms of conic sections follow OpenStax Precalculus analytic geometry.^[1]

OpenStax, Precalculus 2e, Analytic Geometry, https://openstax.org/details/books/precalculus-2e ↩︎