Find the equation for the hyperbola?

Question

The asymptotes are at y= 1/2 x and y= -1/2x. The hyperbolas vertices are at (-3,0) and (3,0). Use the standard form of the hyperbola equation to write an equation for this problem. thanks

Answer 1

Since we don't have any translations here, the general form is:
x^2/a^2 - y^2/b^2 = 1.
a is the position of the vertices; ie a = 3.
The formula for the asymptotes is y = + or - (b/a)x. So b/3 = 1/2, so b = 1.5.
So x^2/3^2 - y^2/1.5^2 = 1.

Answer 2

x²/a² - y²/b² = 1
(b/a)² = 1/4
a = 3 so b² = 9/4

Thus the required parabola is

x²/9 - 4y²/9 = 1
ie x² - 4y² = 9