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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!!
Solution
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