Question

This hyperbola is centered at the origin. Find its equation.

Answer 1

Explanation:

Use these equations.

Equation of a Hyperbola, centered at orgin

`\frac{ {x}^(2) }{ {a}^(2) } - \frac{ {y}^(2) }{ {b}^(2) } = 1`

where a is the major axis

B is the minor axis.

Since a and X are first, the x axis is the major.

This also means the x axis contains the vertices and foci as well.

The equation of vertices when x is major axis is

`(x ±a)`

Since the vertices are (±1,0), and the orgin is (0,0). The length of a is 1.

So a=1.

The equation of foci is

`(x±c)`

Where c is formed by

`{c}^(2) = {a}^(2) + {b}^(2)`

We know that c is 2 since the distance from the foci and center is 2. A is 1, so let find b.

`{2}^(2) = {1}^(2) + {b}^(2)`

`4 = 1 + {b}^(2)`

`3 = {b}^(2)`

`√(3) = {b}`

So our equation is

`\frac{ {x}^(2) }{1 {}^(2) } - \frac{ {y}^(2) }{ (√(3)) {}^(2) } = 1`

`\frac{ {x}^(2) }{1} - \frac{ {y}^(2) }{3} = 1`