Find the standard form of the equation of the hyperbola with the given characteristics and center at the origin.
Vertices: (0,±5); focies: (0,±6)

Question

Find the standard form of the equation of the hyperbola with the given characteristics and center at the origin. ​
Vertices: (0,±5); focies: (0,±6)
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Quizplus · Accepted Answer

The standard form of the equation of a hyperbola centered at the origin is:

(x^2/a^2) - (y^2/b^2) = 1     (for horizontal hyperbola)

or

(y^2/b^2) - (x^2/a^2) = 1     (for vertical hyperbola)

where a is the distance from the center to the vertices, and c is the distance from the center to the foci (using the relationship c^2 = a^2 + b^2).

In this case, we have:

a = 5
c = 6

So, we can find b using:

c^2 = a^2 + b^2

6^2 = 5^2 + b^2

b^2 = 11

b = sqrt(11)

Since the vertices are on the y-axis and the foci are also on the y-axis, we know that this is a vertical hyperbola. Therefore, the standard form of the equation is:

(y^2/11) - (x^2/25) = 1

Which matches Choice C.

Find the Standard Form of the Equation of the Hyperbola