CONIC SECTIONS: THE HYPERBOLA

Example 1:

Find the coordinates of the center, vertices, and foci of the hyperbola given by . The equation is already in standard form where h = 2, k = 5, a = 4 (from denominator of positive term!), and b = 5

Now, we can find the requested information as follows:

Coordinates of the Center (h, k)

(2, 5)

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

(2 + 4, 5) and (2 4, 5)

then (6, 5) and ( 2, 5)

Coordinates of the Foci (h + c, k) and (h c, k), where Since a = 4 and b = 5, then . and 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)