Use the info to find the equation of the hyperbola:
The transverse axis of the hyperbola lies on the line y=–3 and has length 6; the conjugate axis lies on the line x=2 and has length 8.
To find the equation of the hyperbola, we need to determine its center, vertices, and the distance between the center and the foci.
1. Finding the center:
The center of the hyperbola is the point (h, k).
Since the transverse axis lies on the line y = -3, the y-coordinate of the center is -3.
Since the conjugate axis lies on the line x = 2, the x-coordinate of the center is 2.
Therefore, the center of the hyperbola is (2, -3).
2. Finding the vertices:
The vertices of the hyperbola are given by (h +/- a, k), where a is the distance from the center to the vertices along the transverse axis.
The length of the transverse axis is 6, so the distance from the center to the vertices is 6/2 = 3.
Therefore, the vertices of the hyperbola are (2 +/- 3, -3), which can be simplified to (5, -3) and (-1, -3).
3. Finding the distance between the center and the foci:
The distance between the center and the foci along the transverse axis is given by c, where c^2 = a^2 + b^2, and b is the distance from the center to the foci along the conjugate axis.
The length of the conjugate axis is 8, so the distance from the center to the foci is 8/2 = 4.
Using the formula c^2 = a^2 + b^2, we have 4^2 = 3^2 + b^2.
Simplifying, 16 = 9 + b^2.
Therefore, b^2 = 16 - 9 = 7.
Taking the square root of both sides, we find that b = sqrt(7).
So, the distance between the center and the foci is sqrt(7).
Now we have all the necessary information to write the equation of the hyperbola in standard form:
The equation of a hyperbola centered at (h, k) with vertices (h +/- a, k) and distance between the center and foci c is given by:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Plugging in the values we found:
(x - 2)^2 / 3^2 - (y + 3)^2 / (sqrt(7))^2 = 1
Simplifying further, we have:
(x - 2)^2 / 9 - (y + 3)^2 / 7 = 1
Therefore, the equation of the hyperbola is:
(x - 2)^2 / 9 - (y + 3)^2 / 7 = 1