Question
Write the equation of the ellipse with center left-parenthesis 5 comma 3 right-parenthesis, vertex left parenthesis 13 comma 3 right-parenthesis, and focus left-parenthesis 5 plus 2 Start Root 14 End Root comma 3 right-parenthesis period
(1 point)
Responses
Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 8 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 64 End Fraction equals 1
Image with alt text: Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 8 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 64 End Fraction equals 1
Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 8 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 8 End Fraction equals 1
Image with alt text: Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 8 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 8 End Fraction equals 1
Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 64 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 8 End Fraction equals 1
Image with alt text: Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 64 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 8 End Fraction equals 1
Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 64 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 56 End Fraction equals 1
(1 point)
Responses
Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 8 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 64 End Fraction equals 1
Image with alt text: Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 8 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 64 End Fraction equals 1
Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 8 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 8 End Fraction equals 1
Image with alt text: Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 8 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 8 End Fraction equals 1
Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 64 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 8 End Fraction equals 1
Image with alt text: Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 64 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 8 End Fraction equals 1
Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 64 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 56 End Fraction equals 1
Answers
GPT-4o mini
To write the equation of the ellipse, we need the center, the lengths of the semi-major and semi-minor axes, and the orientation of the ellipse. From the problem statement, we have:
- Center: \((h, k) = (5, 3)\)
- Vertex: \((13, 3)\)
- Focus: \((5 + 2\sqrt{14}, 3)\)
**Step 1:** Determine the orientation.
Since the vertex and focus have the same \(y\)-coordinate (3), the ellipse is horizontal.
**Step 2:** Calculate the lengths.
The distance from the center to the vertex gives us \(a\):
- Distance from the center to the vertex:
\[
a = 13 - 5 = 8
\]
The distance from the center to the focus gives us \(c\):
- Distance from the center to the focus:
\[
c = (5 + 2\sqrt{14}) - 5 = 2\sqrt{14}
\]
**Step 3:** Use the relationship between \(a\), \(b\), and \(c\).
For ellipses, the relationship is:
\[
c^2 = a^2 - b^2.
\]
Substituting the known values:
\[
(2\sqrt{14})^2 = 8^2 - b^2.
\]
Calculating \(c^2\) and \(a^2\):
\[
4 \cdot 14 = 56,
\]
\[
64 = 8^2.
\]
Now substitute:
\[
56 = 64 - b^2,
\]
which leads to:
\[
b^2 = 64 - 56 = 8.
\]
**Step 4:** Write the equation of the ellipse.
In general form for a horizontally oriented ellipse centered at \((h, k)\):
\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.
\]
Substituting in the values of \(h\), \(k\), \(a^2 = 64\), and \(b^2 = 8\):
\[
\frac{(x - 5)^2}{64} + \frac{(y - 3)^2}{8} = 1.
\]
**Final Answer:**
\[
\frac{(x - 5)^2}{64} + \frac{(y - 3)^2}{8} = 1.
\]
Therefore, the correct option is:
**Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 64 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 8 End Fraction equals 1**.
- Center: \((h, k) = (5, 3)\)
- Vertex: \((13, 3)\)
- Focus: \((5 + 2\sqrt{14}, 3)\)
**Step 1:** Determine the orientation.
Since the vertex and focus have the same \(y\)-coordinate (3), the ellipse is horizontal.
**Step 2:** Calculate the lengths.
The distance from the center to the vertex gives us \(a\):
- Distance from the center to the vertex:
\[
a = 13 - 5 = 8
\]
The distance from the center to the focus gives us \(c\):
- Distance from the center to the focus:
\[
c = (5 + 2\sqrt{14}) - 5 = 2\sqrt{14}
\]
**Step 3:** Use the relationship between \(a\), \(b\), and \(c\).
For ellipses, the relationship is:
\[
c^2 = a^2 - b^2.
\]
Substituting the known values:
\[
(2\sqrt{14})^2 = 8^2 - b^2.
\]
Calculating \(c^2\) and \(a^2\):
\[
4 \cdot 14 = 56,
\]
\[
64 = 8^2.
\]
Now substitute:
\[
56 = 64 - b^2,
\]
which leads to:
\[
b^2 = 64 - 56 = 8.
\]
**Step 4:** Write the equation of the ellipse.
In general form for a horizontally oriented ellipse centered at \((h, k)\):
\[
\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1.
\]
Substituting in the values of \(h\), \(k\), \(a^2 = 64\), and \(b^2 = 8\):
\[
\frac{(x - 5)^2}{64} + \frac{(y - 3)^2}{8} = 1.
\]
**Final Answer:**
\[
\frac{(x - 5)^2}{64} + \frac{(y - 3)^2}{8} = 1.
\]
Therefore, the correct option is:
**Start Fraction left-parenthesis x minus 5 right-parenthesis squared over 64 End Fraction plus Start Fraction left-parenthesis y minus 3 right-parenthesis squared over 8 End Fraction equals 1**.