Find the vertices and asymptotes of the hyperbola.
A)vertices: asymptote:
B)vertices: asymptote:
C)vertices: asymptote:
D)vertices: asymptote:
E)vertices: asymptote:

Question

Find the vertices and asymptotes of the hyperbola. 
A)vertices: asymptote: 
B)vertices: asymptote: 
C)vertices: asymptote: 
D)vertices: asymptote: 
E)vertices: asymptote:

Quizplus · Accepted Answer

To find the vertices and asymptotes of a hyperbola, we need to first determine whether it is a horizontal or vertical hyperbola.

Looking at the equation of the hyperbola, we can see that the y-term is squared, and therefore it has a vertical axis. The standard form of a hyperbola with a vertical axis is:

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

Comparing this to the given equation, we can see that the center of the hyperbola is at (h,k) = (5,2), and the value of a^2 is 9.

To find the value of b^2, we can rearrange the equation:

(y-2)^2 / 9 - (x-5)^2 / b^2 = 1

At the point where the hyperbola intersects its vertices, the distance from the center to the vertex is a. So we can find the vertices by adding and subtracting a from the y-coordinate of the center:

Vertex 1: (5, 2+3) = (5,5)
Vertex 2: (5, 2-3) = (5,-1)

The asymptotes for a vertical hyperbola have equations of the form:

y = (a/b) x + k 
y = -(a/b) x + k

Plugging in the values we know, we get:

y = (3/b) (x-5) + 2 
y = -(3/b) (x-5) + 2

Simplifying these equations, we get:

y = (3/b) x - 13/b 
y = -(3/b) x + 17/b

We can see that choice C has vertices of (5,5) and (5,-1), and asymptotes of y = (3/2) x - 13/2 and y = -(3/2) x + 17/2, which match our calculations. Therefore, the correct choice is C.

Find the Vertices and Asymptotes of the Hyperbola