Find the vertices, foci, and asymptotes for the hyperbola given by the equation.
A)vertices: , foci: , asymptotes:
B)vertices: , foci: , asymptotes:
C)vertices: , foci: , asymptotes:
D)vertices: , foci: , asymptotes:
E)vertices: , foci: , asymptotes:

Question

Find the vertices, foci, and asymptotes for the hyperbola given by the equation. 
A)vertices: , foci: , asymptotes: 
B)vertices: , foci: , asymptotes: 
C)vertices: , foci: , asymptotes: 
D)vertices: , foci: , asymptotes: 
E)vertices: , foci: , asymptotes:

Quizplus · Accepted Answer

The equation of a hyperbola in standard form is:
$\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1$
where (h,k) is the center of the hyperbola, a is the distance from the center to the vertices, and b is the distance from the center to the co-vertices.

Comparing this to the given equation, we see that the center is at (2,-3), a=4 and b=3.

The vertices are located at (h±a,k), so the vertices here are (6,-3) and (-2,-3). The foci are located at (h±$\sqrt{a^2+b^2}$,k), so the foci here are approximately (5.2,-3) and (-1.2,-3).

The asymptotes can be found using the formula:
$y-k=\pm\frac{b}{a}(x-h)$
Plugging in the values we get two asymptotes:
$y+3=\pm\frac{3}{4}(x-2)$

Therefore, the correct choice is C.

Find the Vertices, Foci, and Asymptotes for the Hyperbola Given