What is the standard form equation of a hyperbola with vertices at (0, 2) and (-10, 2) and foci at (8, 2) and (-18, 2)

Question

What is the standard form equation of a hyperbola with vertices at (0, 2) and (-10, 2) and foci at (8, 2) and (-18, 2)?

 

standard form equation of a hyperbola.

Answer 1

As the vertices are (0, 2) and (-10, 2), the center is at ((0-10)/2, 2) = (-5, 2)

The hyperbola opens up in the x-direction.  The vertices are 0 - -5 = 5 from the center.  Thus, a = 5.

The foci are 8 - -5 = 13 from the center.

The distance to the focus is c.

a² + b² = c²

Thus, b = sqrt(c² - a²) = sqrt(13² - 5²) = sqrt(169 - 25) = sqrt(144) = 12

 

Then, as a = 5, b = 12, and the center is at (-5, 2), the standard form of the hyperbola is:

 

(x + 5)²/5² - (y - 2)²/12² = 1, or (x + 5)²/25 - (y - 2)²/144 = 1