Determine the vertices and foci of the ellipse $\left(x+1\right)^2+\frac{\left(y-2\right)^2}{4}=1$.

• A. Vertices: $\left(-1,4\right),\left(-1,0\right)$Foci: $\left(-1,2+\sqrt3\right),\left(-1,2-\sqrt3\right)$
• B. Vertices: $\left(-1,4\right),\left(-1,0\right)$Foci: $\left(-2,2\right),\left(0,2\right)$
• C. Vertices: $\left(-2,2\right),\left(0,2\right)$Foci: $\left(1,2+\sqrt3\right),\left(1,2-\sqrt3\right)$
• D. Vertices: $\left(-2,2\right),\left(0,2\right)$Foci: $\left(2+\sqrt3,1\right),\left(2-\sqrt3,1\right)$

Answer: (A)

Verified step by step guidance

1. Identify the standard form of the ellipse equation: \( \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \), where \((h, k)\) is the center, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.
2. Rewrite the given equation \((x+1)^2 + \frac{(y-2)^2}{4} = 1\) in standard form. Here, \(h = -1\), \(k = 2\), \(a = 2\), and \(b = 1\).
3. Determine the vertices by using the formula \((h, k \pm a)\) for a vertical ellipse. Substitute \(h = -1\), \(k = 2\), and \(a = 2\) to find the vertices.
4. Calculate the foci using the formula \((h, k \pm c)\), where \(c = \sqrt{a^2 - b^2}\). Compute \(c\) using \(a = 2\) and \(b = 1\).
5. Substitute \(h = -1\), \(k = 2\), and \(c\) into the foci formula to find the coordinates of the foci.