Write an equation for the hyperbola with vertices (7,5),(-5,5); foci (11,5),(-9,5)
To write the equation for the hyperbola with given vertices and foci, we need to determine the center, a, b, and c values.
1. Find the center: The center of the hyperbola can be calculated by taking the average of the vertices' x-coordinates and y-coordinates. In this case, the x-coordinate of the center is (7 + (-5)) / 2 = 1/2, and the y-coordinate remains 5. So, the center of the hyperbola is (1/2, 5).
2. Find the a value: The distance between the center and either vertex is referred to as "a." In this case, the distance between the center (1/2, 5) and either vertex (7, 5) is 7 - 1/2 = 13/2. So, a = 13/2.
3. Find the c value: The distance between the center and either focus is called "c." In this case, the distance between the center (1/2, 5) and either focus (11, 5) is 11 - 1/2 = 21/2. So, c = 21/2.
4. Find the b value: To find "b," we can use the formula b² = c² - a², where b represents the distance from the center to either the vertex or foci along the transverse axis. In this case, a = 13/2 and c = 21/2. Thus, b² = (21/2)² - (13/2)² gives us b² = 441/4 - 169/4 = 272/4 = 68. Taking the square root of both sides, we get b = √68 = 2√17.
Now that we have the necessary values, we can write the equation of the hyperbola with center (1/2, 5):
(x - h)² (y - k)²
--------- - --------- = 1
a² b²
Replacing h and k with the coordinates of the center, and substituting the values of a and b, we get:
(x - 1/2)² (y - 5)²
------------- - ------------- = 1
(13/2)² (2√17)²
Simplifying further:
4(x - 1/2)² (y - 5)²
------------ - ------------- = 1
169 68
Therefore, the equation of the hyperbola is:
4(x - 1/2)² (y - 5)²
------------ - ------------- = 1
169 68