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.