Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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Question
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### Initial Knowledge Check
#### Task:
Find an equation of the parabola with focus \((7, -2)\) and directrix \(x = -9\).
#### Explanation:
You're given a problem to find the equation of a parabola. The parabola's defining components—the focus and the directrix—are provided:
- **Focus**: \((7, -2)\)
- **Directrix**: \(x = -9\)
To solve this, you'll use the formula for a parabola based on the distance from a point to a line and a point:
The equation of a parabola with a vertical axis of symmetry is given by:
\[ (x - h)^2 = 4p(y - k) \]
Conversely, the equation for a parabola with a horizontal axis (opening left or right) is:
\[ (y - k)^2 = 4p(x - h) \]
Where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus or directrix.
In this task, since the directrix is vertical (\(x = -9\)), the parabola opens horizontally.
- Determine the vertex location using the midpoint formula between the focus and directrix.
- Calculate \(p\), which is half the distance between the directrix and focus point.
After finding the vertex and \(p\), substitute these into the horizontal parabola equation:
\[ (y - k)^2 = 4p(x - h) \]
---
Note: Ensure you understand the process of finding the vertex and the parabola's axis to approach similar problems confidently.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac3be618-b645-4bd8-855b-808ff3fc18d4%2Ffe92c188-025b-4200-9df9-058e3a0b00f3%2Flzhu26o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:---
### Initial Knowledge Check
#### Task:
Find an equation of the parabola with focus \((7, -2)\) and directrix \(x = -9\).
#### Explanation:
You're given a problem to find the equation of a parabola. The parabola's defining components—the focus and the directrix—are provided:
- **Focus**: \((7, -2)\)
- **Directrix**: \(x = -9\)
To solve this, you'll use the formula for a parabola based on the distance from a point to a line and a point:
The equation of a parabola with a vertical axis of symmetry is given by:
\[ (x - h)^2 = 4p(y - k) \]
Conversely, the equation for a parabola with a horizontal axis (opening left or right) is:
\[ (y - k)^2 = 4p(x - h) \]
Where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus or directrix.
In this task, since the directrix is vertical (\(x = -9\)), the parabola opens horizontally.
- Determine the vertex location using the midpoint formula between the focus and directrix.
- Calculate \(p\), which is half the distance between the directrix and focus point.
After finding the vertex and \(p\), substitute these into the horizontal parabola equation:
\[ (y - k)^2 = 4p(x - h) \]
---
Note: Ensure you understand the process of finding the vertex and the parabola's axis to approach similar problems confidently.
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