To write the equation of the ellipse, we can use the standard form for a vertical ellipse, which is given by:
\[ \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \]
where \((h, k)\) is the center of the ellipse, \(a\) is the semi-major axis, and \(b\) is the semi-minor axis.
From the information provided:
- The center of the ellipse is \((-1, 2)\).
- The top vertex is \((-1, 5)\) and the lower vertex is \((-1, -1)\).
The distance from the center \((-1, 2)\) to either vertex (the semi-major axis) is:
\[ a = 5 - 2 = 3 \quad (or) \quad a = 2 - (-1) = 3 \]
So, \(a = 3\) and therefore \(a^2 = 9\).
The vertices are only vertically aligned along the line \(x = -1\), which indicates the semi-minor axis will be horizontal. The distance to the endpoints along the horizontal (using the information and structure you provided) appears to be \(b = 2\) (as the option with these values indicates correctly), making \(b^2 = 4\).
Putting these values into the equation, we have:
\[ \frac{(x + 1)^2}{4} + \frac{(y - 2)^2}{9} = 1 \]
Thus, the equation of the ellipse is:
\[ \frac{(x + 1)^2}{4} + \frac{(y - 2)^2}{9} = 1 \]
The correct response is:
start fraction left parenthesis x plus 1 right parenthesis squared over 4 end fraction plus start fraction left parenthesis y minus 2 right parenthesis squared over 9 end fraction equals 1.