Write the equation of a hyperbola..?

Question

1.) with given vertices (-2,0) and (2,0) and with asymptotes at y= +- 2x ? 

2.)with given center (0,0) , vertex (3,0) , and focus (6,0)?

Answer 1

2).

vertices : (3, 0) and (- 3, 0), focus:(± 6, 0), and center:(h, k) = (0, 0).

The x - coordinates are different in vertices, indicating a horizontal axis and horizontal hyperbola:
(x - h)²/a² -(y - k)²/b² = 1.

length of axis = 2a  = v(- 3 - 3)² + (0 - 0)² = 6

a = 3.

Distance between foci.

2c =v(- 6 - 6)² + (0 - 0)² =v(- 12)²= 12.

c = 12/2 = 6

Find the value of b:

b² = c² - a²

    = 6² - 3²

    = 36 - 9

     = 27.


b² = 27.

Equation of the hyperbola

(x - 0)²/3² - (y - 0)²/27 = 1

x²/9 - y²/27 = 1.

Answer 2

1).

vertices : (- 2, 0) and (2, 0),  and asymtotes are : y = ± 2x.

The x - coordinates are different in vertices, indicating a horizontal axis and horizontal hyperbola:
(x - h)²/a² -(y - k)²/b² = 1.

length of axis = 2a  = v(- 2 - 2)² + (0 - 0)² = 4.

a = 2.

Let the center be (0, 0).

Then, asymtotes = ± (b/a)x = ± 2x.

b/a = 2

b/2 = 2

b = 4.

Equation of the hyperbola

(x - 0)²/2² - (y - 0)²/4² = 1

x²/4 - y²/16 = 1.