Question
Write the equation of the ellipse with center at left parenthesis 2 comma 1 right parenthesis, one vertex at left parenthesis 2 comma negative 4 right parenthesis, and one focus at left parenthesis 2 comma negative 2 right parenthesis.
(1 point)
Responses
start fraction left parenthesis x minus 2 right parenthesis squared over 9 end fraction plus start fraction left parenthesis y minus 1 right parenthesis squared over 25 end fraction equals 1
Image with alt text: start fraction left parenthesis x minus 2 right parenthesis squared over 9 end fraction plus start fraction left parenthesis y minus 1 right parenthesis squared over 25 end fraction equals 1
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
Image with alt 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
start fraction left parenthesis x minus 2 right parenthesis squared over 25 end fraction plus start fraction left parenthesis y minus 1 right parenthesis squared over 9 end fraction equals 1
Image with alt text: start fraction left parenthesis x minus 2 right parenthesis squared over 25 end fraction plus start fraction left parenthesis y minus 1 right parenthesis squared over 9 end fraction equals 1
start fraction left parenthesis x minus 2 right parenthesis squared over 25 end fraction plus start fraction left parenthesis y minus 1 right parenthesis squared over 16 end fraction equals 1
(1 point)
Responses
start fraction left parenthesis x minus 2 right parenthesis squared over 9 end fraction plus start fraction left parenthesis y minus 1 right parenthesis squared over 25 end fraction equals 1
Image with alt text: start fraction left parenthesis x minus 2 right parenthesis squared over 9 end fraction plus start fraction left parenthesis y minus 1 right parenthesis squared over 25 end fraction equals 1
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
Image with alt 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
start fraction left parenthesis x minus 2 right parenthesis squared over 25 end fraction plus start fraction left parenthesis y minus 1 right parenthesis squared over 9 end fraction equals 1
Image with alt text: start fraction left parenthesis x minus 2 right parenthesis squared over 25 end fraction plus start fraction left parenthesis y minus 1 right parenthesis squared over 9 end fraction equals 1
start fraction left parenthesis x minus 2 right parenthesis squared over 25 end fraction plus start fraction left parenthesis y minus 1 right parenthesis squared over 16 end fraction equals 1
Answers
GPT-4o mini
To find the equation of the ellipse with the given characteristics, we first identify important details:
1. **Center**: The center of the ellipse is at (2, 1).
2. **Vertex**: One vertex is at (2, -4).
3. **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:
\[
\text{start fraction left parenthesis x minus 2 right parenthesis squared over 16 end fraction plus start fraction left parenthesis y minus 1 right parenthesis squared over 25 end fraction equals 1}
\]
1. **Center**: The center of the ellipse is at (2, 1).
2. **Vertex**: One vertex is at (2, -4).
3. **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:
\[
\text{start fraction left parenthesis x minus 2 right parenthesis squared over 16 end fraction plus start fraction left parenthesis y minus 1 right parenthesis squared over 25 end fraction equals 1}
\]