Find the standard form of the equation of the hyperbola with the given characteristics.
Vertices: (6, 0),(10, 0); foci: (0, 0), (12, 0)

Question

Find the standard form of the equation of the hyperbola with the given characteristics. ​
Vertices: (6, 0),(10, 0); foci: (0, 0), (12, 0)
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Quizplus · Accepted Answer

The center of the hyperbola is the midpoint between the vertices, which is ((6+10)/2,0) = (8,0).

The distance between the center and each vertex is a, so a = 2.

The distance between the center and each focus is c, so c = 6.

The formula for the standard form of a hyperbola centered at (h,k) with vertices at (h+a,k) and (h-a,k), and with foci at (h+c,k) and (h-c,k) is:

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

Plugging in the values we know, we get:

((x-8)^2 / 2^2) - ((y-0)^2 / b^2) = 1

Simplifying:

(x-8)^2 / 4 - y^2 / b^2 = 1

We need to find b. Using the formula b^2 = c^2 - a^2, we get:

b^2 = 6^2 - 2^2 = 32

So the final answer is:

(x-8)^2 / 4 - y^2 / 32 = 1

which is choice C.

Find the Standard Form of the Equation of the Hyperbola