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A hyperbola has vertices (±5, 0) and one focus (6, 0). What is the standard-form equation of the hyperbola?
Given that the hyperbola has vertices at (±5, 0) and one focus at (6, 0), we can determine the key components needed to find the standard form equation of the hyperbola:
Center: The center of the hyperbola is the midpoint between the vertices, which is at (0,0).
a: The distance between the center and one of the vertices is 5 units, so a = 5.
c: The distance between the center and one of the foci is 6 units, so c = 6.
Now we can use the formula for a hyperbola: c^2 = a^2 + b^2 to find the value of b.
Plugging in the values, we get:
6^2 = 5^2 + b^2
36 = 25 + b^2
b^2 = 11
b = √11
Now we can write the standard-form equation of the hyperbola:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Plugging in the values of a, b, and the center (h,k), we have:
(x - 0)^2 / 5^2 - (y - 0)^2 / √11^2 = 1
x^2 / 25 - y^2 / 11 = 1
Therefore, the standard-form equation of the hyperbola is:
x^2 / 25 - y^2 / 11 = 1