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...
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### 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

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