# How do you find the center and vertices and sketch the hyperbola 4y^2-x^2=1?

Jan 31, 2017

See explanation and graph.

#### Explanation:

TIn the standard form the equation is

${y}^{2} / {\left(\frac{1}{2}\right)}^{2} - {x}^{2} / {1}^{1} = 1$

Center C is the origin (0, 0)

Major axis : y-axis, x = 0.

Transverse axis : x-axis, y = 0.

Semi major axis $a = \frac{1}{2}$

Semi transverse axis $b = 1$

Eccentricity : $e = \sqrt{1 + {a}^{2} / {b}^{2}} = \frac{\sqrt{5}}{2}$

Vertices A and A' on major axis x = 0 : $\left(0 , \pm a\right) = \left(0 , \pm \frac{1}{2}\right)$-

Foci S and S' on major axis : (0, +-ae)=(0, +-sqrt5)

The asymptotes : $4 {y}^{2} - {x}^{2} = 0$, giving $y = \pm \frac{x}{\sqrt{2}}$.

The Socratic graph is inserted.

graph{(4y^2-x^2-1)(4y^2-x^2)((y-sqrt5)^2+x^2-.01)((y+sqrt5)^2+x^2-.01)((y-.5)^2+x^2-.01)((y+.5)^2+x^2-.01)=0 [-5, 5, -2.5, 2.5]}