To find the equation of the ellipse with the given characteristics, we first identify important details:
- Center: The center of the ellipse is at (2, 1).
- Vertex: One vertex is at (2, -4).
- Focus: One focus is at (2, -2).
Since the center and both the vertex and focus have the same x-coordinate (2), this indicates that the major axis of the ellipse is vertical.
Key Parameters
- Center: \( (h, k) = (2, 1) \)
- Vertex (V): \( (h, k - a) = (2, -4) \)
From the center (2, 1) to the vertex (2, -4), the distance is: \[ a = 1 - (-4) = 5 \]
- Focus (F): \( (h, k - c) = (2, -2) \)
From the center (2, 1) to the focus (2, -2), the distance is: \[ c = 1 - (-2) = 3 \]
Relationship
For an ellipse, \( c^2 = a^2 - b^2 \). We know:
- \( a = 5 \) ⇒ \( a^2 = 25 \)
- \( c = 3 \) ⇒ \( c^2 = 9 \)
Substituting into the relationship: \[ 9 = 25 - b^2 \implies b^2 = 25 - 9 = 16 \implies b = 4 \]
Equation of the Ellipse
The standard form of the equation of a vertical ellipse is: \[ \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1 \]
Substituting \( h = 2 \), \( k = 1 \), \( a^2 = 25 \), and \( b^2 = 16 \): \[ \frac{(x - 2)^2}{16} + \frac{(y - 1)^2}{25} = 1 \]
Thus, the equation of the ellipse is: \[ \frac{(x - 2)^2}{16} + \frac{(y - 1)^2}{25} = 1 \]
Conclusion
So, the correct response is: \[ \text{start fraction left parenthesis x minus 2 right parenthesis squared over 16 end fraction plus start fraction left parenthesis y minus 1 right parenthesis squared over 25 end fraction equals 1} \]