Question

How do you write the equation of a hyperbola given foci at (1, 5) and (7, 5) and with vertices at (2, 5) and (6, 5)?

Answer 1

Use the equation of hyperbola and `a^2 +b^2 = c^2`

Explanation:

The foci and the vertices all lie along the horizontal line y=5, so the hyperbola opens left/right. This implies that the x term goes first and the y term is subtracted.

The center is half way between the vertices at (4.5) = (h,k).

The vertices are 2 units from the center, so a=2. `a^2 = 4`.

The foci are 3 units from center, so c =3.

Using `a^2 + b^2 = c^2` gives `b^2 = 5`.

Plugging into `((x-h)^2)/a^2 - ((y-k)^2)/b^2 = 1`

gives `((x-4)^2)/4 - ((y-5)^2)/5 = 1`