Find the equation of the hyperbola if:

Question

a). the asymptotes are y=0.5x and y=−4−0.5x, and one of the vertices is (−4, 0).

b). it has x-orientation, with vertices (2, 0) and (−2, 0), and it goes through the point (4, 5sqrt(3))

PLEASE HELP AND THANK YOU IF YOU DO!!

Answer 1

For a) The center of the hyperbola is at the intersection of the two asymptotes. This is the point (-4,-2)

Since the vertex at (-4,0) is above this point, we have y orientation. From this, the equation will be of the form

-a (x + 4)² + b (y + 2)² = 1 for some a and b

Plugging in (-4,0) shows that b = 1/4

Demanding that y = x/2 be an asymptote shows that a = 1/16 , so finally

-(1/16) (x +4)² + (1/4) (y +2)² = 1

For b) The center is at (0,0) so the equation will be

a x² - b y² = 1 for some a and b.

Plugging in (2,0) shows that a = 1/4 Plugging in (4, 5 sqrt(3)) leads to b = 1/25 , so

(1/4) x² - (1/25) y² = 1