Calculus: Early Transcendentals
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ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![### Problem 8: Conic Section Equation
**Problem Statement:**
Find the equation of the conic whose focus is \((4, -5)\) and directrix is \(x = -2\).
**Explanation:**
To determine the type of conic section and its equation, we use the definition of a conic section: the set of all points \((x, y)\) such that the ratio of the distance from \((x, y)\) to the focus to the distance from \((x, y)\) to the directrix is constant.
In this case:
- **Focus:** \((4, -5)\)
- **Directrix:** \(x = -2\)
The standard form of the equation for a conic section is determined by the given information and adjusting for different conic shapes (ellipses, parabolas, hyperbolas).
#### Step-by-Step Solution:
1. **Identify the conic type:**
Since we have one focus and one directrix, this indicates that the conic is a parabola.
2. **Parabola Equation (Vertex Form):**
A parabola's equation with focus \((h, k)\) and directrix \(x = h - p\) (if the directrix is vertical) can be written as:
\[
(x - h)^2 = 4p(y - k)
\]
Here, the vertex \((h, k)\) of the parabola is the midpoint between the focus and the directrix.
3. **Calculate \(h\), \(k\), and \(p\):**
- \(h = 4\)
- \(k = -5\)
- Distance from the focus to the directrix: \(4 - (-2) = 6\)
- Half of this distance is the vertex distance from the directrix:
\(p = \frac{6}{2} = 3\)
4. **Formulate the equation:**
Since the directrix \(x = -2\) is vertical, the equation for the parabola is formed using this value of \(p\):
\[
(x - 4)^2 = 4(3)(y + 5)
\]
Simplifying:
\[
(x - 4)^2 = 12(y + 5)
\]](https://content.bartleby.com/qna-images/question/5c941f29-ddef-4885-9d8b-99e8246804cb/6b9aff9a-51b8-4001-b113-bb4a4568a928/jt2n4nh.jpeg)
Transcribed Image Text:### Problem 8: Conic Section Equation
**Problem Statement:**
Find the equation of the conic whose focus is \((4, -5)\) and directrix is \(x = -2\).
**Explanation:**
To determine the type of conic section and its equation, we use the definition of a conic section: the set of all points \((x, y)\) such that the ratio of the distance from \((x, y)\) to the focus to the distance from \((x, y)\) to the directrix is constant.
In this case:
- **Focus:** \((4, -5)\)
- **Directrix:** \(x = -2\)
The standard form of the equation for a conic section is determined by the given information and adjusting for different conic shapes (ellipses, parabolas, hyperbolas).
#### Step-by-Step Solution:
1. **Identify the conic type:**
Since we have one focus and one directrix, this indicates that the conic is a parabola.
2. **Parabola Equation (Vertex Form):**
A parabola's equation with focus \((h, k)\) and directrix \(x = h - p\) (if the directrix is vertical) can be written as:
\[
(x - h)^2 = 4p(y - k)
\]
Here, the vertex \((h, k)\) of the parabola is the midpoint between the focus and the directrix.
3. **Calculate \(h\), \(k\), and \(p\):**
- \(h = 4\)
- \(k = -5\)
- Distance from the focus to the directrix: \(4 - (-2) = 6\)
- Half of this distance is the vertex distance from the directrix:
\(p = \frac{6}{2} = 3\)
4. **Formulate the equation:**
Since the directrix \(x = -2\) is vertical, the equation for the parabola is formed using this value of \(p\):
\[
(x - 4)^2 = 4(3)(y + 5)
\]
Simplifying:
\[
(x - 4)^2 = 12(y + 5)
\]
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