Find an equation for the hyperbola with vertices (2,0) and (-2,0) and asymptote y=1/2x

Question

(A) 4x^2-y^2=4
(B) x^2-4y^2=4
(C) x^2-y^2=4
(D) x^2+4y^2=4
Thank you very much!

Answer 1

The vertices of hyperbola is (2, 0) and (- 2, 0) and asymptote y = 1/2 x.

The standard form of equation of hyperbola with center at the origin (where a and b are not equals to 0) is x^2/a^2 - y^2/b^2 = 1 (Transverse axis is horizontal) or y^2/a^2 - x^2/b^2 = 1 (Transverse axis is vertical).

The x - coordinates of the  vertices points are 2 and - 2.

The value of a = 2, because the vertices are two units from the center.

Because the transverse axis is horizontal, the asymptotes are of the forms y = (b/a) x and y = - (b/a) x.

The asymptote y = 1/2 x is comparison with y = (b/a) x.

b/a = 1/2 ------> b = 1 and a = 2.

The hyperbola equation is x^2/2^2 - y^2/1^2 = 1.

x^2/4 - y^2 = 1.

(x^2 - 4y^2)/4 = 1.

x^2 - 4y^2 = 4.

The option B is correct choice.