Write the equation of the hyperbola with the following info:
The hyperbola has vertices (–2, 9) and (–2, 3) and foci (–2, 13) and (–2, –1);
To write the equation of a hyperbola, we need the following information:
1. The coordinates of the center (h, k)
2. The distances from the center to the vertices (a)
3. The distances from the center to the foci (c)
From the given information, we can determine:
1. The center is (-2, 6), which is the midpoint between the vertices (–2, 9) and (–2, 3).
- The x-coordinate of the center remains the same because the hyperbola is symmetric about the y-axis.
- The y-coordinate of the center is the average of the corresponding y-coordinates of the vertices.
2. The distance from the center to the vertices (a) can be found as the absolute difference between the y-coordinate of the center and either vertex, which in this case is |6 - 9| = 3.
3. The distance from the center to the foci (c) is calculated similarly. It is the absolute difference between the y-coordinate of the center and either focus, which is |6 - 13| = 7.
Now that we have the necessary information, we can write the equation of the hyperbola in standard form:
- For a hyperbola with a horizontal transverse axis:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
- For a hyperbola with a vertical transverse axis:
(y - k)^2 / a^2 - (x - h)^2 / b^2 = 1
Since the y-axis is the transverse axis in this case, the equation of the hyperbola is:
(y - 6)^2 / 3^2 - (x + 2)^2 / b^2 = 1
Now, we need to find the value of "b" to complete the equation. The relationship between a, b, and c in a hyperbola is given by:
c^2 = a^2 + b^2
Substituting the known values:
7^2 = 3^2 + b^2
49 = 9 + b^2
b^2 = 49 - 9
b^2 = 40
Finally, we can substitute the value of b^2 into the equation to get the final equation of the hyperbola:
(y - 6)^2 / 3^2 - (x + 2)^2 / 40 = 1