Math · Precalculus · Concept

Conic sections: parabolas, ellipses, hyperbolas

Circles, ellipses, parabolas and hyperbolas are the curves a plane cuts from a double cone, and each has a second-degree equation in x and y. Completing the square turns the general equation into a standard form that shows the center and radius of a circle, and the vertices, foci and asymptotes of the other conics.

Four curves, one family

A plane cutting a double cone makes a circle, an ellipse, a parabola or a hyperbola, depending on its tilt. Algebraically, each is the graph of Ax² + Cy² + Dx + Ey + F = 0: equal A and C give a circle, A and C of the same sign an ellipse, exactly one of them zero a parabola, and opposite signs a hyperbola. A few equations of this form give only a point, a pair of lines or no graph at all.

Standard forms centered at the origin
Conic	Equation	Key features
Circle	x² + y² = r²	Radius r
Ellipse	x²/a² + y²/b² = 1	Vertices (±a, 0); c² = a² − b²
Parabola	y² = 4px	Focus (p, 0); directrix x = −p
Hyperbola	x²/a² − y²/b² = 1	Asymptotes y = ±(b/a)x; c² = a² + b²

Circles

A circle is every point at distance r from its center (h, k). Squaring the distance formula gives its equation.

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

Completing the square

An equation such as x² + y² − 6x + 4y − 12 = 0 hides its center. Group the x-terms and the y-terms, then add the square of half of each linear coefficient to both sides.

(x^{2} - 6 x + 9) + (y^{2} + 4 y + 4) = 12 + 9 + 4

\begin{aligned} (x^{2} - 6 x + 9) \\ + (y^{2} + 4 y + 4) = 12 + 9 + 4 \end{aligned}

Ellipses

An ellipse is every point whose distances to two foci add to the same total, 2a. In standard form the larger denominator is a², and it lies under the variable of the major axis. The foci sit c units from the center along that axis.

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, c^{2} = a^{2} - b^{2}

Parabolas

A parabola is every point equally far from a focus and a line called the directrix. With its vertex at the origin, y² = 4px opens to the right when p > 0, with focus (p, 0) and directrix x = −p; x² = 4py opens upward in the same way.

y^{2} = 4 p x, x^{2} = 4 p y

Hyperbolas

A hyperbola is every point whose distances to two foci differ by the same amount, 2a. The minus sign in its equation splits it into two branches, which approach the asymptotes y = ±(b/a)x. Here c² = a² + b², so the foci lie beyond the vertices.

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1, c^{2} = a^{2} + b^{2}

Moving the center

Replacing x with x − h and y with y − k moves any of these curves so that its center or vertex is at (h, k), just as the same change shifts the graph of a function.

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

Common mistakes

Adding the completing-the-square numbers to one side only.
Reading the center’s signs straight from the equation: (x − 3)² + (y + 2)² = 25 has center (3, −2).
Giving r² as the radius: (x − 3)² + (y + 2)² = 25 has radius 5.
Using c² = a² − b² for a hyperbola: for a hyperbola, c² = a² + b².
Assuming the x-denominator is always a²: in an ellipse a² is the larger denominator, and it sets the direction of the major axis.

Key terms

Conic section: A curve that a plane cuts from a double cone: a circle, an ellipse, a parabola or a hyperbola. Each is the graph of a second-degree equation in x and y.
Equation of a circle: The equation (x − h)² + (y − k)² = r², satisfied by exactly the points at distance r from the center (h, k). It is the distance formula, squared.
Completing the square: Rewriting x² + bx as (x + b/2)² − (b/2)² by adding and subtracting (b/2)². It turns a quadratic equation into a square equal to a constant, and a quadratic function into vertex form.
Ellipse: The set of points whose distances to two fixed foci add to a constant 2a. Centered at the origin with a horizontal major axis, its equation is x²/a² + y²/b² = 1 with a ≥ b > 0, and its foci are at (±c, 0), where c² = a² − b².
Parabola: The set of points equally far from a fixed point, the focus, and a fixed line, the directrix. With its vertex at the origin, x² = 4py opens upward if p > 0, with focus (0, p) and directrix y = −p. The graph of every quadratic function is a parabola.
Hyperbola: The set of points whose distances to two fixed foci differ by a constant 2a. Centered at the origin and opening left and right, its equation is x²/a² − y²/b² = 1; its foci are at (±c, 0), where c² = a² + b², and its asymptotes are y = ±(b/a)x.
Focus and directrix: The fixed point and fixed line that define a parabola: every point on the curve is as far from the focus as from the directrix. An ellipse or a hyperbola has two foci, and the sum or the difference of the distances to them is constant.

Work through an example

Find the center and radius of the circle x² + y² − 6x + 4y − 12 = 0.

Find the center and radius of a circle →

Find the vertices and foci of an ellipse →

Find the vertices and asymptotes of a hyperbola →

Find the focus and directrix of a parabola →

Find where a line meets a circle →

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