Find the equation in standard form for the hyperbola that satisfies the given conditions: transverse axis endpoints (-2,-2) and (-2,7), slope of one asymptote 4/3.

I found the distance of the transverse axis to be 9. For the formula, I have a=9/2. I need help finding h, k, and b.

Thanks!

To find the equation of the hyperbola in standard form, we need to determine the values of h, k, a, and b.

Let's start by identifying the center of the hyperbola. Since the transverse axis has endpoints (-2, -2) and (-2, 7), the center of the hyperbola is the midpoint of these points.

To find the midpoint, we average the x-coordinates and y-coordinates separately:
x-coordinate of the center = (-2 + -2) / 2 = -4 / 2 = -2
y-coordinate of the center = (-2 +7) / 2 = 5 / 2 = 2.5

So, the center of the hyperbola is (h, k) = (-2, 2.5).

Next, we can find the value of a. The distance between the center and each endpoint of the transverse axis represents half of the length of the transverse axis. In this case, the distance is 9.

Therefore, a = 9 / 2.

Now, we need to find the value of b. The asymptotes of the hyperbola have a slope of 4/3. The slope of the asymptotes is given by b / a, where b is half of the length of the conjugate axis.

Slope of the asymptotes = b / a = (4/3).

We know that b^2 = a^2 * (slope of asymptotes)^2.
Let's simplify this equation using the values we already know:

b^2 = (9 / 2)^2 * (4/3)^2
b^2 = 81/4 * 16/9
b^2 = 16/1
b^2 = 16

To find b, we take the square root of both sides:

b = ±√16 = ±4

Now that we have h, k, a, and b, we can write the equation of the hyperbola in standard form:

(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1

Plugging in the values, we get:

(x + 2)^2 / (9/2)^2 - (y - 2.5)^2 / 4^2 = 1

Simplifying further (by squaring the denominator):

(x + 2)^2 / 81/4 - (y - 2.5)^2 / 16 = 1

Multiplying through by the common denominator (81/4):

4(x + 2)^2 - 81(y - 2.5)^2 / 81 = 1

Simplifying:

4(x + 2)^2 - 81(y - 2.5)^2 = 81

Therefore, the equation of the hyperbola in standard form that satisfies the given conditions is:

4(x + 2)^2 - 81(y - 2.5)^2 = 81