### 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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Answered: 8. Find the equation of the conic whose…

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Calculus

8. Find the equation of the conic whose focus is (4, -5) and directrix is x = -2.

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 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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