Determine the vertices and foci of the hyperbola $\frac{y^2}{4}-x^2=1$.

• A. Vertices: $\left(2,0\right),\left(-2,0\right)$Foci: $\left(\sqrt5,0\right),\left(-\sqrt5,0\right)$
• B. Vertices: $\left(0,2\right),\left(0,-2\right)$Foci: $\left(0,\sqrt5\right),\left(0,-\sqrt5\right)$
• C. Vertices: $\left(1,0\right),\left(-1,0\right)$Foci: $\left(5,0\right),\left(-5,0\right)$
• D. Vertices: $\left(0,1\right),\left(0,-1\right)$Foci: $\left(0,5\right),\left(0,-5\right)$

Answer: (B)

Verified step by step guidance

1. Identify the standard form of the hyperbola equation. The given equation is \( \frac{y^2}{4} - x^2 = 1 \), which is in the form \( \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1 \).
2. Determine the values of \( a^2 \) and \( b^2 \). From the equation, \( a^2 = 4 \) and \( b^2 = 1 \). Therefore, \( a = 2 \) and \( b = 1 \).
3. Identify the orientation of the hyperbola. Since the \( y^2 \) term is positive, the hyperbola opens vertically.
4. Calculate the vertices of the hyperbola. For a vertical hyperbola, the vertices are at \( (0, \pm a) \). Thus, the vertices are \( (0, 2) \) and \( (0, -2) \).
5. Calculate the foci of the hyperbola. The distance to the foci \( c \) is given by \( c = \sqrt{a^2 + b^2} = \sqrt{4 + 1} = \sqrt{5} \). Therefore, the foci are at \( (0, \pm \sqrt{5}) \).