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## Equations of Hyperbolas

#### 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/a2 - (y-k)2/b2 =1, vertical: (y-k) 2/a2 -(x-h)2/b2 =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: x2/a2 -y2/b2 =1, vertical: y2/a2 -x2/b2 =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, c2=a2+b2 (±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=((x1+x2) / 2 ,(y1 + y2)/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  #### 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.

• Quiz 1 - Find an equation for the hyperbola with the center at (4, 14), vertex at (0, 14), and focus at (18, 14).
• Quiz 2 - The center is in the middle of the foci.
• Quiz 3 - Write the standard equation of each Hyperbola: x2 / 9 – (y+4)2 / 36 = 1