Example 2:

Find the coordinates of the center, vertices, and foci of the hyperbola given by .

First, we will convert the equation of the hyperbola to standard form. Since there is a positive y²-term raised to the second power we know that this hyperbola is of the form .

NOTE: The center (h, k) of this hyperbola is at (0, 0) because there is only a y²-term and an x²-term. Therefore, we can write

where h = 0, k = 0, a = 5 (from denominator of positive term!), and b = 4

Now, we can find the requested information as follows:

Coordinates of the Center (h, k)

(0, 0)

Coordinates of the Vertices (h, k + a) and (h, k a)

(0, 0 + 5) and (0, 0 5)

then (0, 5) and (0, 5)

Coordinates of the Foci (h, k + c) and (h, k c), where

Since a = 5 and b = 4, then .

and

then and

Example 3:

Find the coordinates of the center, vertices, foci, and the equations of the asymptotes of the hyperbola given by .

First, we will convert the equation of the hyperbola to standard form.

It is either or , where a = b or a b

Let's divide both sides of the equation by 12 to get

and

or

We can now write the standard form of the given hyperbolic equation as follows:

where h = 1, k = 0, a = 2 (from denominator of positive term!), and b =

Now, we can find the requested information as follows:

Coordinates of the Center (h, k)

(1, 0)

Coordinates of the Vertices (h, k + a) and (h, k a)

(1, 0 + 2) and (1, 0 2)

(1, 2) and (1, 2)

Coordinates of the Foci (h, k + c) and (h, k c), where

Since a = 2 and b = , then .

and

and

Example 4:

Find the coordinates of the center, vertices, and foci of the hyperbola given by .

First, we will convert the equation of the hyperbola to standard form. Since there is a positive x²-term raised to the second power we know that this hyperbola is of the form.

The given equation is almost in standard form. All we have to do is give the x²-term a denominator of 1 as follows:

where h = 1, k = 2, a = 1 (from denominator of positive term!), and b = 3

Now, we can find the requested information as follows:

Coordinates of the Center (h, k)

(1, 2)

Coordinates of the Vertices (h + a, k) and (h a, k)

(1 + 1, 2) and (1 1, 2)

then (2, 2) and (0, 2)

Coordinates of the Foci (h + c, k) and (h c, k), where

Since a = 1 and b = 3, then .

(1 + 2, 2) and (1 2, 2)

then (3, 2) and (1, 2)