A graph of a vertical ellipse is shown.A graph of a vertical ellipse is shown. A dashed vertical line segment indicates major axis. The top vertex is labeled left-parenthesis negative 1 comma 5 right-parenthesis, the center is labeled left-parenthesis negative 1 comma 2 right-parenthesis and the lower vertex is labeled left-parenthesis negative 1 comma negative 1 right-parenthesis.

Question
Use the image to answer the question.
Write the equation of the ellipse shown in the graph.
(1 point)
Responses

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
Image with alt text: 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

start fraction left parenthesis x plus 1 right parenthesis squared over 9 end fraction plus start fraction left parenthesis y minus 2 right parenthesis squared over 4 end fraction equals 1
Image with alt text: start fraction left parenthesis x plus 1 right parenthesis squared over 9 end fraction plus start fraction left parenthesis y minus 2 right parenthesis squared over 4 end fraction equals 1

start fraction left parenthesis x minus 1 right parenthesis squared over 4 end fraction plus start fraction left parenthesis y plus 2 right parenthesis squared over 9 end fraction equals 1
Image with alt text: start fraction left parenthesis x minus 1 right parenthesis squared over 4 end fraction plus start fraction left parenthesis y plus 2 right parenthesis squared over 9 end fraction equals 1

start fraction left parenthesis x minus 1 right parenthesis squared over 9 end fraction plus start fraction left parenthesis y plus 2 right parenthesis squared over 4 end fraction equals 1

1 answer

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.