find the value of 4p

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...
icon
Related questions
icon
Concept explainers
Topic Video
Question
### Problem Statement

Given the focus and directrix, find the value of \(\frac{1}{4p}\).

- **Focus**: \((4, 5)\)
- **Directrix**: \(y = 11\)

### Options

- a) \(-\frac{1}{12}\)
- b) \(\frac{1}{24}\)
- c) \(\frac{1}{12}\)
- d) \(-\frac{1}{24}\)

### Explanation

To solve this problem, typically, the parabola in question is vertical since the directrix is a horizontal line \(y = 11\). The vertex is midway between the focus \((4, 5)\) and the directrix \(y = 11\).

The distance between the focus and directrix is \(11 - 5 = 6\). Therefore, the vertex is at \(y = 8\) (average of 5 and 11), and the complete relationship in terms of \(p\) (distance from vertex to focus or directrix) can be found.

Then, compute \(\frac{1}{4p}\) based on this information, selecting from the given options.
Transcribed Image Text:### Problem Statement Given the focus and directrix, find the value of \(\frac{1}{4p}\). - **Focus**: \((4, 5)\) - **Directrix**: \(y = 11\) ### Options - a) \(-\frac{1}{12}\) - b) \(\frac{1}{24}\) - c) \(\frac{1}{12}\) - d) \(-\frac{1}{24}\) ### Explanation To solve this problem, typically, the parabola in question is vertical since the directrix is a horizontal line \(y = 11\). The vertex is midway between the focus \((4, 5)\) and the directrix \(y = 11\). The distance between the focus and directrix is \(11 - 5 = 6\). Therefore, the vertex is at \(y = 8\) (average of 5 and 11), and the complete relationship in terms of \(p\) (distance from vertex to focus or directrix) can be found. Then, compute \(\frac{1}{4p}\) based on this information, selecting from the given options.
Expert Solution
Step 1

Focus: (4, 5)

Directrix: y = 11

 

As we know, Focus=h, k+p and Directrix: y=k-p

Here, (h, k) is center.

Therefore,

k+p=5            .............1And,k-p=11          .............2

steps

Step by step

Solved in 2 steps

Blurred answer
Knowledge Booster
Angles, Arcs, and Chords and Tangents
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning