In a hyperbola the absolute value of the difference of distances from any point on a hyperbola to the foci is constant, so .

Since we want to determine the equation for a hyperbola with a vertical transverse axis

centered at the origin, use the distance formula .

Let .

From the defination of a hyperbola .

Subustiute the values of in .

.

.

Isolate one radical and then square both sides.

Square the obtained equation on both sides.

Divide with on both sides.

Graph of the hyperbola .

(1).Draw the coordinate plane.

(2).Graph the hyperbola.

The general form of hyperbola is derived as .