Find the foci and the equation of the hyperbola with the vertices (0,±1) and asymptote y = 2x

NOTE:

i. The hyperbola  has foci (±c, 0), where c2 = a2 + b2,vertices (±a, 0), and asymptotes .

ii. The hyperbola  has foci (0, ±c), where c2 = a2 + b2,vertices (0, ±a), and asymptotes .

Solution:  From Note (ii), and given information in the example, we can see that a=1 and a/b = 2. Thus b=a/2 = 1/2 and c2 = a2 + b2 = 5/4. Te foci are and the equation of the hyperbola is given by y2 - 4x2 = 1.