Question

How do you write an equation for the hyperbola with center (0,0), vertex (-2,0) and focus (4,0)?

Answer 1

Please see the explanation.

Explanation:

The given, center, vertex, and focus share the same y coordinate, 0, ,therefore, the standard form for the equation of this type of hyperbola is the one corresponding to the Horizontal Transverse Axis type:

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

where `(h, k)` is the center, `a` is distance from the center to the vertex, and `b` affects the distance, `c`, from the center to the focus as determined by the equation `c^2 = a^2 + b^2`.

Substitute the center `(0, 0)` into the standard form:

`(x - 0)^2/a^2 - (y - 0)^2/b^2 = 1`

We know that `a = 2`, because one of the vertices a distance of 2 from the center. Substitute 2 for a into the equation:

`(x - 0)^2/2^2 - (y - 0)^2/b^2 = 1`

Observing that the focus is a distance of 4 from the center we set `c^2 = 16` in the equation `c^2 = a^2 + b^2`:

`16 = a^2 + b^2`

Substitute 4 for `a^2`

`16 = 4 + b^2`

Solve for b:

`b= sqrt(12)`

Substitute `sqrt(12)` for b into the equation:

`(x - 0)^2/2^2 - (y - 0)^2/(sqrt(12))^2 = 1`