Unit 4 · Topic 4.6

4.6 Conic Sections

Parabolas, ellipses and hyperbolas are the conic sections, curves you get by slicing a cone with a plane. Each has a standard equation in x and y that shows its center or vertex, its size, and the direction it opens.

Key terms

conic section
parabola
ellipse
hyperbola
vertex and center

Parabolas

A parabola with vertex (h, k) that opens up or down: y − k = a(x − h)². It opens up if a > 0 and down if a < 0.

A parabola with vertex (h, k) that opens left or right: x − h = a(y − k)². It opens right if a > 0 and left if a < 0.

The squared variable tells you the orientation. If x is squared, the parabola opens vertically; if y is squared, it opens horizontally.

Ellipses and circles

An ellipse centered at (h, k) with horizontal radius a and vertical radius b: (x − h)²/a² + (y − k)²/b² = 1, where a and b are positive.

The ellipse reaches a units left and right of the center and b units above and below it. If a > b it's wider than it is tall; if b > a it's taller.

A circle is the special case a = b: (x − h)² + (y − k)² = r².

Hyperbolas

A hyperbola centered at (h, k) that opens left and right: (x − h)²/a² − (y − k)²/b² = 1. Its vertices are (h ± a, k).

One that opens up and down: (y − k)²/b² − (x − h)²/a² = 1. Its vertices are (h, k ± b).

Either way, the asymptotes are the lines y − k = ±(b/a)(x − h). Draw the rectangle reaching a units left and right and b units up and down from the center; the asymptotes run through its corners.

The positive squared term tells you the direction: if the x-term is positive, it opens left and right.

Getting to standard form

If an equation isn't in standard form, group the x-terms and the y-terms and complete the square for each. Then divide so the right side is 1.

This course only uses conics with horizontal and vertical lines of symmetry, so no xy terms. It also focuses on what you need to sketch: centers, vertices, radii and asymptotes. Foci, directrices and eccentricity aren't required.

You can often identify the conic before doing any algebra. If x² and y² both appear with the same sign, it's an ellipse (a circle if their coefficients are equal). If they have opposite signs, it's a hyperbola. If only one variable is squared, it's a parabola.

Worked examples

Try each one yourself first, then open the solution.

Example 1
Completing the square for an ellipse
Write 4x² + 9y² − 16x + 54y + 61 = 0 in standard form and describe the ellipse.
Show the solutionHide the solution
1. Step 1: Group: 4(x² − 4x) + 9(y² + 6y) = −61.
2. Step 2: Complete the squares: 4(x² − 4x + 4) + 9(y² + 6y + 9) = −61 + 16 + 81 = 36. (Adding 4 inside the first group adds 4 · 4 = 16, and adding 9 inside the second adds 9 · 9 = 81.)
3. Step 3: So 4(x − 2)² + 9(y + 3)² = 36. Divide by 36: (x − 2)²/9 + (y + 3)²/4 = 1.
Answer: (x − 2)²/9 + (y + 3)²/4 = 1: center (2, −3), horizontal radius 3, vertical radius 2.
Example 2
Reading a hyperbola
Describe (y − 1)²/16 − (x + 2)²/9 = 1.
Show the solutionHide the solution
1. Step 1: The y-term is positive, so it opens up and down. Center (−2, 1).
2. Step 2: b² = 16, so b = 4: vertices (−2, 1 + 4) = (−2, 5) and (−2, 1 − 4) = (−2, −3).
3. Step 3: a² = 9, so a = 3. Asymptotes: y − 1 = ±(4/3)(x + 2).
Answer: Opens up and down; center (−2, 1); vertices (−2, 5) and (−2, −3); asymptotes y − 1 = ±(4/3)(x + 2).
Example 3
Trap: which way does it open?
Which way does x − 3 = −2(y + 1)² open, and where is its vertex?
Show the solutionHide the solution
1. Step 1: The trap is to see the negative coefficient and say “opens down.”
2. Step 2: Here y is squared, so the parabola opens horizontally. The coefficient −2 is negative, so it opens to the left.
3. Step 3: Vertex: (h, k) = (3, −1).
Answer: It opens to the left, with vertex (3, −1).

Common mistakes

Mixing up h and k signs: (x + 2)² means h = −2.
Forgetting to add the same amount to both sides when completing the square, especially when a coefficient is factored out.
Using a and b as diameters instead of radii in an ellipse.
Judging a parabola's direction from the sign alone without checking which variable is squared.

On the exam

Unit 4 is not on the AP Precalculus Exam, so you'll see this topic on class tests rather than in May.
Expect to identify the type of conic from its equation, put it in standard form, and give its center or vertex, radii or vertices, and asymptotes for a hyperbola.

Connected topics

Videos

AP Precalculus – 4.6A Conic Sections: Parabolas
The AlgebrosWatch on YouTube (opens in a new tab)
AP Precalculus – 4.6B Conic Sections: Ellipses
The AlgebrosWatch on YouTube (opens in a new tab)
Introduction to conic sections | Conic sections | Algebra II | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)
AP PreCalculus - 4.6.2 Conics (Hyperbolas)
COACH MEIDENBAUERWatch on YouTube (opens in a new tab)
Graphing Conic Sections Part 4: Hyperbolas
Professor Dave ExplainsWatch on YouTube (opens in a new tab)