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 and insert minus signs 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 (0, 2) and (2, 2)
Coordinates of the Foci (h + c, k) and (h c, k), where
Since a = 1 and b = 3, then .
and