Find the standard form of the equation of the hyperbola with the given characteristics. vertices: (-6, -2), (-2, -2)
Asymptotes:

Question

Quizplus · Accepted Answer

The center of the hyperbola can be found by finding the midpoint between the vertices, which is ((-6-2)/2, (-2-2)/2) = (-4,-2). 
The distance between the vertices is 4 units, which is also the length of the transverse axis. Thus, a = 2.
The slope of the asymptotes is given by ±(b/a), where b is the distance between the center and each branch of the hyperbola. Since the hyperbola is symmetric, each branch has the same distance from the center, which can be found using the Pythagorean theorem with the vertices and the center: 
sqrt((-6+4)^2+(-2+2)^2) = sqrt(4) = 2. 
Therefore, b=2sqrt(3).
The asymptotes have the equation y+2 = ±(2sqrt(3)/2)(x+4), which simplifies to y = ±sqrt(3)(x+4) - 2.
The standard form of the equation of a hyperbola centered at (h,k) with transverse axis length 2a and asymptotes y = ±(b/a)(x-h) + k is 
(x-h)^2/a^2 - (y-k)^2/b^2 = 1.
Substituting the given values, we have 
(x+4)^2/4 - (y+2)^2/(12) = 1, 
which is equivalent to 11eb80b1_b110_d932_a291_3f7fc0601fec_TB8633_11.

Find the Standard Form of the Equation of the Hyperbola