Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Problem Statement:**
Find an equation for the conic that satisfies the given conditions.
**Conic Type:** Ellipse
**Given Conditions:**
- **Foci:** \((0, 2)\), \((0, 10)\)
- **Vertices:** \((0, 0)\), \((0, 12)\)
**Solution:**
To determine the equation of the ellipse, note that the foci and vertices lie on the vertical line \(x = 0\), indicating a vertically oriented ellipse. The center \((h, k)\) is the midpoint of the vertices \((0, 0)\) and \((0, 12)\). Thus, the center is:
\[ (h, k) = (0, 6) \]
The distance from the center to each vertex \(a\) is:
\[ a = 6 \]
The distance between the foci is 8, so the distance from the center to each focus \(c\) is:
\[ c = 4 \]
Using the relationship \(c^2 = a^2 - b^2\):
\[ 4^2 = 6^2 - b^2 \]
\[ 16 = 36 - b^2 \]
\[ b^2 = 20 \]
The standard form of the equation for a vertically oriented ellipse is:
\[
\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1
\]
Substituting the known values:
\[
\frac{(x-0)^2}{20} + \frac{(y-6)^2}{36} = 1
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fabb75337-a8ce-4ff9-8c07-7d58c277bdaa%2F78629b5f-3a5d-48ca-8892-56aa3a4c8cfb%2F153wpx_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find an equation for the conic that satisfies the given conditions.
**Conic Type:** Ellipse
**Given Conditions:**
- **Foci:** \((0, 2)\), \((0, 10)\)
- **Vertices:** \((0, 0)\), \((0, 12)\)
**Solution:**
To determine the equation of the ellipse, note that the foci and vertices lie on the vertical line \(x = 0\), indicating a vertically oriented ellipse. The center \((h, k)\) is the midpoint of the vertices \((0, 0)\) and \((0, 12)\). Thus, the center is:
\[ (h, k) = (0, 6) \]
The distance from the center to each vertex \(a\) is:
\[ a = 6 \]
The distance between the foci is 8, so the distance from the center to each focus \(c\) is:
\[ c = 4 \]
Using the relationship \(c^2 = a^2 - b^2\):
\[ 4^2 = 6^2 - b^2 \]
\[ 16 = 36 - b^2 \]
\[ b^2 = 20 \]
The standard form of the equation for a vertically oriented ellipse is:
\[
\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1
\]
Substituting the known values:
\[
\frac{(x-0)^2}{20} + \frac{(y-6)^2}{36} = 1
\]
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