## Equations of Hyperbolas

#### Aligned To Common Core Standard:

**Expressing Properties** - HSG-GPE.A.3

How to find the Equation of a Hyperbola?
In analytic geometry, there are several different types of curves, and the most common ones include hyperbolas and ellipses.
Hyperbola is a smooth curve lying in a plane. When a right circular cone intersects with a plane at a specified angle, which cuts both halves of the cone, it produces two unbounded curves, which we call the hyperbolas. Both curves are a reflection of each other and are not connected.
A hyperbola may occur vertically or horizontally;
horizontal: (x-h)^{2}/a^{2} - (y-k)^{2}/b^{2} =1, vertical: (y-k) ^{2}/a^{2} -(x-h)^{2}/b^{2} =1
These are the two patterns of a hyperbola, and you can determine the equation of a hyperbola using these patterns. These are used when the center is not the origin. When the origin is the center, the pattern becomes;
horizontal: x^{2}/a^{2} -y^{2}/b^{2} =1, vertical: y^{2}/a^{2} -x^{2}/b^{2} =1
Here,
(h,k) are the coordinates of the center point. (±a,0) are the vertices. (0,±b) are the coordinates of co-vertices. a and b are connected via the formula, c^{2}=a^{2}+b^{2}
(±c,0) are the coordinates of the foci. The distance between two vertices is equal to 2a. The center can be calculated using the midpoint formula using the two vertices.
Midpoint=((x_{1}+x_{2}) / 2 ,(y_{1} + y_{2})/2)
Distance between the center and the focus is given by ae. e here is the eccentricity. By substituting all these values in the equation, you can get the equation of a hyperbola. This series of worksheets and lessons will help students learn to write and understand equations for hyperbolas.

### Printable Worksheets And Lessons

- Finding Hyperbolas
Step-by-step Lesson - What is the equation based on the center,
vertex, and focus?

- Guided Lesson
- Question number three mixes it up a bit to make sure you are reading
everything.

- Guided Lesson Explanation
- There is a general formula listed on the first page that applies
to all questions this way I didn't need to write it over and over.

- Practice Worksheet
- Go hyperbola equation crazy with these.

- Matching Worksheet
- We play equation tennis. Go back and forth until you find matches
for all your work.

- Hyperbolas - Find the Foci and
Vertices Worksheet Five Pack - Some of these equations are bit
difficult to interpret.

- Hyperbolas Equations Worksheet
Five Pack - More work like the previous pack.

#### Homework Sheets

Given a few measures and location, we want you to find the equation of the hyperbolas for us.

- Homework 1 - The point that the hyperbola is focused (pointed) on is referred to as the center.
- Homework 2 - When we use a coordinate system, the recognizable point that is on the branch closest to the center of the hyperbola is called the vertex.
- Homework 3 - The foci reside inside each branch of the hyperbola.

#### Practice Worksheets

These sheets get progressively harder.

- Practice 1 - Find an equation of the hyperbola with x-intercepts at x = –12 and x =6, and foci at (–16, 0) and (10, 0).
- Practice 2 - If we look at the foci, we will see that they are side-by-side. This indicates that branches of the hyperbola follow this lead. This also means that the center, foci, and vertices are on a line that is parallel to the x-axis.
- Practice 3 - The center resides on the x-axis. This means that the xintercepts have to also be vertices for the hyperbola.

#### Math Skill Quizzes

If you handled the homework and practice in stride, these are pretty straight forward for you.