Find an equation for the hyperbola whose transverse axis is horizontal, with center at (6, -3) and passes through the points (13, 3) and (1, -3).

Question

Find an equation for the hyperbola whose transverse axis is horizontal, with center at (6, -3) and passes through the points (13, 3) and (1, -3).

Please help, thank you!

Answer 1

The Standard Equation for Hyperbola if the transverse axis is horizontal is:

(x-h)²/a² -(y-k)²/b² = 1

where C(h,k) = Center of hyperbola

Given:

C(h,k) = (6, -3)

Through the points (13,3),(1,-3)

Using the center (6,-3) and (1,-3) to plug in the standard equation, We have:

(1-6)²/a² -(-3-(-3))²/b² = 1

(-5)²/a² - (-3+3)²/b² = 1

25/a² = 1

25 = a²

a = 5

Now let's solve for b using the value of a=5, center (6,-3) and (13,3):

(13-6)²/5² - (3+3)²/b² = 1

7²/5² - 6²/b² = 1

49/25 - 36/b² = 1

49/25 - 1 = 36/b²

49/25 - 25/25 = 36/b²

24/25 = 36/b²

b² = (36)(25/24) = (3)(25/2)

b² = 75/2 = 37.5

Therefore the equation of the hyperbola is:

(x-6)²/25 - (y+3)²/37.5 = 1