Find an equation for the hyperbola that has the vertices 1 0

Solution

The center is at ((-1+1)/2, 0) = (0, 0)

Thus, as (1,0) is a distance 1 from (0, 0) in the x direction, the equation of the hyperbola is of the form

x² / 1 - y ²/a² = 1

Equations of this form have slopes of the form ±a. In this case, a = -3

Thus, our function is of the form

x² /1 - y ²/3² = 1, or

x² /1 - y ²/9 = 1

If you wish to derive this, note that, for x² / 1 - y ²/a² = 1,

y ²/a² = x² / 1 - 1, or

y² = a² (x² - 1) or

y = ±a(x² - 1)

As, asymptotically, (x² - 1) = |x|,

y = ±ax.

Thus, we again see that a = -3 in the form of the hyperbola.