# Writing the Equation of Hyperbolas

We can write the equation of a hyperbola by following these steps:

1. Identify the center point (h, k)

2. Identify a and c

3. Use the formula c

^{2}= a

^{2}+ b

^{2}to find b (or b

^{2})

4. Plug h, k, a, and b into the correct pattern.

5. Simplify

Sometimes you will be given a graph and other times you might just be told some information.

Let's try a few.

1. Find the equation of a hyperbola whose vertices are at (-1, -1) and (-1, 7) and whose foci are at (-1, 8) and (-1, -2).

2. Find the equation of this hyperbola:
To start, let's graph the information we have:

We can tell that it is a vertical hyperbola. Let's find our center point next and mark it. If we want, we can also draw in a rough hyperbola just to make it easier to visualize:

The center point is (-1, 3). To find a, we'll count from the center to either vertex. a = 4. To find c, we'll count from the center to either focus. c = 5

We'll use the formula c

And simplify:

We can tell that it is a vertical hyperbola. Let's find our center point next and mark it. If we want, we can also draw in a rough hyperbola just to make it easier to visualize:

The center point is (-1, 3). To find a, we'll count from the center to either vertex. a = 4. To find c, we'll count from the center to either focus. c = 5

We'll use the formula c

^{2}= a^{2}+ b^{2}to find b. To do that, we'll sub in a = 4 and c = 5 then solve for b.
c

5

25 = 16 + b

9 = b

b = 3

We have all our information: h = -1, k = 3, a = 4, b = 3. Since it's a vertical hyperbola, we'll choose that formula and substitute in our information.^{2}= a^{2}+ b^{2}5

^{2}= 4^{2}+ b^{2}25 = 16 + b

^{2}9 = b

^{2}We need to take the square root.b = 3

And simplify:

We can tell that it is a horizontal hyperbola. Let's find our center point next and mark it.k

The center point is (1, 2). To find a, we'll count from the center to either vertex. a = 2. To find c, we'll count from the center to either focus. c = 6

We'll use the formula c

^{2}= a

^{2}+ b

^{2}to find b. To do that, we'll sub in a = 2 and c = 6 then solve for b.

c

6

36 = 4 + b

32 = b

To find b, we would need to take the square root, but it won't come out evenly. That's okay, though, because the pattern needs b^{2}= a^{2}+ b^{2}6

^{2}= 2^{2}+ b^{2}36 = 4 + b

^{2}32 = b

^{2}^{2}, so we can just substitute in 32 for b

^{2}.

We have all our information: h = 1, k = 2, a = 2, b

^{2}= 32 . Since it's a horizontal hyperbola, we'll choose that formula and substitute in our information.

And simplify:

**Practice:**Find the equation of each parabola:

1) Vertices: (2, 1) and (2, -5) Foci: (2, 3) and (2, -7)

2) Vertices: (0, 1) and (6, 1) Foci: (-1, 1) and (7, 1)

3) Vertices: (1, 0) and (3, 0) Foci: (-1, 0) and (5, 0)

2) Vertices: (0, 1) and (6, 1) Foci: (-1, 1) and (7, 1)

3) Vertices: (1, 0) and (3, 0) Foci: (-1, 0) and (5, 0)

**Answers:**

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