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How to find the equation of a hyperbola?

Question:

The center of a hyperbola is at the origin of the coordinate. The eccentricity of the hyperbola is 1.25. The coordinate of one focus is (5, 0). Find the equation of the hyperbola.

Solution:

Because the center of the hyperbola is at the origin, and its one focus lies in the x-axis, so, the equation of the hyperbola is x2/a2 - y2/b2 = 1, in which a > 0 and b > 0. Because the coordinate of one focus is (5, 0), so, c = 5

Because the eccentricity of the hyperbola is 1.25, so, from the definition e = c/a, so, a = c/e = 5/1.25 = 4

Now, we know a and c, we can find b. For hyperbola, c2 = a2 + b2. So, b2 = c2 - a2 = 52 - 42 = 25 - 16 = 9 = 32. So, b = 3.

So, the equation of the hyperbola is: x2/16 - y2/9 = 1

The graph of the hyperbola is in this video. The graph also shows the vertices of the hyperbola and two asymptotes of the hyperbola. Look the video for mor details.