Topic 7 of 8

Conic sections

Step 13 · Algebra 2 topics

Recognize circles, parabolas, ellipses and hyperbolas from their equations and their focus definitions, and find their centers, vertices, foci and asymptotes.

Circles and parabolas

Slicing a double cone at different angles gives the conic sections: circles, ellipses, parabolas and hyperbolas. Each has an equation and a definition by distances.

A circle is every point at distance r from a center. A parabola is every point the same distance from a point (the focus) as from a line (the directrix).

(x - h)^{2} + (y - k)^{2} = r^{2} \qquad y = \frac{1}{4p}x^{2}

Example: A parabola's focus

Find the focus and directrix of $y = \frac{1}{8}x^{2}$ .

1 ÷ 4p = 1 ÷ 8, so p = 2.
The vertex is (0, 0), the focus is (0, 2) and the directrix is y = −2.

AnswerFocus (0, 2), directrix y = −2

y = x² ÷ 8: every point on it is as far from the focus (0, 2) as from the dashed line y = −2.

Ellipses

An ellipse is every point whose distances to two foci add to the same total. Centered at the origin:

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

It reaches a along the x-axis and b along the y-axis. When a is bigger than b, the foci are c from the center along the x-axis, where c² = a² − b².

Example: Reading an ellipse

Describe $\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$ .

a = 5 and b = 3: it crosses the x-axis at ±5 and the y-axis at ±3.
c² = 25 − 9 = 16, so the foci are at (±4, 0).

Answer10 wide, 6 high, with foci at (±4, 0)

The ellipse x²/25 + y²/9 = 1 with its foci at (−4, 0) and (4, 0).

Hyperbolas

A hyperbola is every point whose distances to two foci differ by the same amount. Centered at the origin and opening sideways:

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

Its vertices are (±a, 0), its asymptotes are the lines y = ±(b ÷ a)x, and its foci are c from the center, where c² = a² + b².

Example: Reading a hyperbola

Describe $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$ .

a = 4 and b = 3: the vertices are (±4, 0).
The asymptotes are $y = \pm\frac{3}{4}x$ .
c² = 16 + 9 = 25, so the foci are (±5, 0).

AnswerVertices (±4, 0), foci (±5, 0)

x²/16 − y²/9 = 1: two branches that run ever closer to the dashed asymptotes.

Equation	Shape
x² + y² = 16	circle
x² + 4y² = 16	ellipse
x² − 4y² = 16	hyperbola
y = x² + 4	parabola

Try it yourself

Work each one on paper first, then open the answer to check.

Find the center and radius of (x − 3)² + (y + 1)² = 49.
Show answer
Center (3, −1), radius 7
Find the focus of $y = \frac{1}{12}x^{2}$ .
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(0, 3)
Where does $\frac{x^{2}}{36} + \frac{y^{2}}{4} = 1$ cross the x-axis?
Show answer
At x = 6 and x = −6
Find the foci of $\frac{x^{2}}{25} + \frac{y^{2}}{16} = 1$ .
Show answer
(±3, 0)
Find the vertices of $\frac{x^{2}}{9} - \frac{y^{2}}{16} = 1$ .
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(±3, 0)
What shape is 4x² + 9y² = 36?
Show answer
An ellipse