Write an equation of the hyperbola given that the center is at (2, -3), the vertices are at (2, 3) and
(2, - 9), and the foci are at (2, ± 2√10).
use what you know about hyperbolas.
Since the center and vertices are bot at x=2, the axis is vertical. So, given the center, we have
(y+3)^2/a^2 - (x-2)^2/b^2 = 1
However, you have garbled the foci. Since the center is at y = -3, the foci cannot be symmetric about the x-axis.
Fix that, and then, knowing that
a = 6
and c = the real distance to the foci,
and b^2 = c^2-a^2
then you can write the equation.
To write the equation of a hyperbola, we need some information such as the center, the vertices, and the foci. The general equation for a hyperbola with center (h, k) is:
((x - h)^2)/a^2 - ((y - k)^2)/b^2 = 1
where a is the distance from the center to the vertices, and c is the distance from the center to the foci, related by the equation c^2 = a^2 + b^2.
Given that the center is at (2, -3) and the vertices are at (2, 3) and (2, -9), we can determine the value of a. The distance between the center and either of the vertices is the value of a:
a = distance from (2, -3) to (2, 3) = 6
We also know that the foci are at (2, ± 2√10), which means c = 2√10.
Using the equation c^2 = a^2 + b^2, we can find b:
c^2 = a^2 + b^2
(2√10)^2 = 6^2 + b^2
40 = 36 + b^2
b^2 = 40 - 36
b^2 = 4
Now, we have all the necessary information to write the equation of the hyperbola:
((x - 2)^2)/6^2 - ((y + 3)^2)/4^2 = 1
Simplifying, we get:
(x - 2)^2/36 - (y + 3)^2/16 = 1
Therefore, the equation of the hyperbola is ((x - 2)^2)/36 - ((y + 3)^2)/16 = 1.