4.6 Conic Sections

TLDR

Conic sections are the parabola, ellipse (with the circle as a special case), and hyperbola, and in AP Precalculus you represent each one analytically using its standard equation with center or vertex at $(h, k)$. Once you recognize which equation matches which shape, you can pull out key features like center, vertex, radii, and asymptotes.

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Why This Matters for the AP Precalculus Exam

Unit 4 topics, including conic sections, are not assessed on the AP Precalculus exam. The exam covers Units 1, 2, and 3, since those are the topics used for college credit and placement. Schools may still teach this material because it builds skills you will use in calculus and other math and science courses.

So treat this guide as preparation for the bigger picture. The thinking you practice here, recognizing a function or relation from its equation and reading off key features, is exactly the kind of reasoning the exam rewards in the units that are tested. Conic sections also connect directly to implicit equations and parametric forms, which round out your understanding of how curves in the plane can be described.

Key Takeaways

• A parabola with vertex $(h, k)$ is written as $y - k = a(x - h)^2$ when it opens up or down, or as $x - h = a(y - k)^2$ when it opens left or right, with $a \neq 0$.
• An ellipse centered at $(h, k)$ has the form $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$, where $a$ is the horizontal radius and $b$ is the vertical radius.
• A circle is just an ellipse with $a = b$.
• A hyperbola centered at $(h, k)$ is $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ when it opens left and right, or $\frac{(y - k)^2}{b^2} - \frac{(x - h)^2}{a^2} = 1$ when it opens up and down.
• The asymptotes of a hyperbola are $y - k = \pm \frac{b}{a}(x - h)$.
• The sign between the squared terms tells you the shape: a plus sign means an ellipse, a minus sign means a hyperbola, and only one squared term means a parabola.

Conic Sections Overview

Conic sections are curves you get when a plane slices through a cone. The four types are the circle, ellipse, parabola, and hyperbola. In AP Precalculus you focus on conics with horizontal or vertical symmetry, meaning they are not tilted, and you represent each one with a standard equation centered or vertexed at $(h, k)$.

The fastest way to identify a conic from its equation is to look at the squared terms:

• One variable is squared and the other is not, then it is a parabola.
• Both variables are squared and added, then it is an ellipse (a circle if the radii are equal).
• Both variables are squared and subtracted, then it is a hyperbola.

Parabola

A parabola is the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. The vertex is the turning point of the curve.

With vertex $(h, k)$ and a nonzero constant $a$, a parabola is represented as:

• $y - k = a(x - h)^2$ if it opens up or down.
• $x - h = a(y - k)^2$ if it opens left or right.

The sign of $a$ tells you direction. For an up or down parabola, positive $a$ opens up and negative $a$ opens down. For a left or right parabola, positive $a$ opens right and negative $a$ opens left.

Ellipse and Circle

An ellipse is defined by the property that the sum of the distances from any point on the curve to two fixed points (the foci) is constant. An ellipse centered at $(h, k)$ is:

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

Here $a$ is the horizontal radius and $b$ is the vertical radius. The center is $(h, k)$, and the curve stretches $a$ units left and right and $b$ units up and down from the center.

A circle is the special case where $a = b$. With radius $r$, a circle centered at $(h, k)$ can be written as:

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

The foci of an ellipse lie on the longer axis, and the focal length $c$ satisfies $c^2 = a^2 - b^2$ (using the larger of the two radii as $a$). This focus relationship is useful background, but the standard equation above is what you build directly from the center and radii.

Hyperbola

A hyperbola centered at $(h, k)$ with horizontal and vertical lines of symmetry is represented as:

• $\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$ if it opens left and right.
• $\frac{(y - k)^2}{b^2} - \frac{(x - h)^2}{a^2} = 1$ if it opens up and down.

The center is $(h, k)$, and the asymptotes are:

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

The asymptotes are the diagonal lines the branches approach but never touch. The term that comes first and is positive tells you which direction the hyperbola opens.

How to Use This on the AP Precalculus Exam

Even though Unit 4 is not tested, the recognition and feature-reading skills here carry over to graded units. Practice these moves so they become automatic.

Identifying the Conic

Look at the squared terms first.

• Only one squared term means parabola.
• Two squared terms added means ellipse or circle.
• Two squared terms subtracted means hyperbola.

Then check signs and coefficients to find orientation and key features.

Reading Off Key Features

Once you match the equation to its shape, pull out features directly:

• Center or vertex: read $(h, k)$ from the equation. Remember the signs flip, so $(x - 3)^2$ gives $h = 3$.
• Ellipse radii: $a$ is the horizontal radius and $b$ is the vertical radius.
• Hyperbola asymptotes: use $y - k = \pm \frac{b}{a}(x - h)$.

Common Trap

The orientation of a hyperbola depends on which squared term is positive, not which letter sits under it. For $\frac{(y - k)^2}{b^2} - \frac{(x - h)^2}{a^2} = 1$, the $y$ term is positive, so it opens up and down.

Common Misconceptions

• Mixing up plus and minus. Added squared terms give an ellipse, and subtracted squared terms give a hyperbola. The operation between the terms decides the shape.
• Forgetting the sign flip on $(h, k)$. In $(x - h)^2$, the center coordinate is $+h$, not $-h$. So $(x + 2)^2$ means $h = -2$.
• Assuming $a$ is always bigger than $b$ in an ellipse. In the standard form used here, $a$ is the horizontal radius and $b$ is the vertical radius, so either one can be larger depending on the shape.
• Thinking a parabola needs both variables squared. A parabola has exactly one squared variable. If both are squared, it is an ellipse or hyperbola.
• Confusing hyperbola orientation. The positive squared term determines the opening direction, not the variable that appears first by habit.
• Treating the circle as separate from the ellipse. A circle is just an ellipse with equal radii, $a = b$.

Practice Problems

1. What type of conic section is represented by the equation $(x - 2)^2 - (y + 3)^2 = 1$?

a) Circle b) Ellipse c) Parabola d) Hyperbola

1. What type of conic section is represented by the equation $\frac{(x - 4)^2}{9} + \frac{(y + 2)^2}{4} = 1$?

a) Circle b) Ellipse c) Parabola d) Hyperbola

Answers

#1 d) Hyperbola

The equation has two squared terms that are subtracted and set equal to 1, which is the standard form of a hyperbola. The center is $(2, -3)$, and since the $x$ term is positive, it opens left and right.

#2 b) Ellipse

The equation has two squared terms that are added and set equal to 1, which is the standard form of an ellipse. The center is $(4, -2)$, with a horizontal radius of 3 and a vertical radius of 2. Since the horizontal radius is larger, the ellipse is stretched horizontally.

Related AP Precalculus Guides

• 4.10 Matrices
• 4.12 Linear Transformations and Matrices
• 4.5 Implicitly Defined Functions
• 4.9 Vector-Valued Functions
• 4.13 Matrices as Functions
• 4.4 Parametrically Defined Circles and Lines