/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 $$\text {Give the focus, directr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 10: Problem 21

$$\text {Give the focus, directrix, and axis of symmetry for each parabola.}$$ $$x^{2}=24 y$$

Short Answer

Expert verified
Focus: \( (0, 6) \), Directrix: \( y = -6 \), Axis of symmetry: \( x = 0 \)

Step by step solution

01

Identify the form of the given equation

The given equation is \( x^{2} = 24y \). Compare this to the standard form of a parabola that opens upwards or downwards, which is \( x^{2} = 4py \).
02

Determine the value of \( p \)

From the standard form \( x^{2} = 4py \), equate \( 4p \) with 24 to find the value of \( p \). Thus, \( 4p = 24 \implies p = 6 \).
03

Find the focus

The focus of the parabola \( x^{2} = 4py \) is at the point \( (0, p) \). Given that \( p = 6 \), the focus is at \( (0, 6) \).
04

Find the directrix

The directrix of the parabola \( x^{2} = 4py \) is the line \( y = -p \). Given that \( p = 6 \), the directrix is \( y = -6 \).
05

Identify the axis of symmetry

For the parabola \( x^{2} = 4py \), the axis of symmetry is the y-axis, or the line \( x = 0 \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

focus of a parabola
The focus of a parabola is a special point located inside the curve of the parabola. It plays a significant role in defining the shape and position of the parabola.
For the parabola given by the equation \( x^{2} = 24y \), we can compare this to the standard form \( x^{2} = 4py \).
By doing so, we determine that \( 4p = 24 \, \ p = 6 \. \)The focus is then located at the point \( (0, p) \. \) So in this case, the focus is at \( (0, 6) \).
This means that if you were to draw this parabola on a graph, the focus would be 6 units above the origin, right at the point (0, 6). The focus helps determine how 鈥渟teep鈥� or 鈥渇lat鈥� the parabola appears.
directrix of a parabola
The directrix of a parabola is a line that is perpendicular to the axis of symmetry and serves as a mirror line for the parabolic curve.
In the equation format we are dealing with, \( x^{2} = 4py \), the directrix is given by \ y = -p \.
So, for the given problem where \ p = 6 \, the directrix becomes \ y = -6 \.
This directrix line is always positioned below the focus and acts as a boundary to guide the shape of the parabola.
Every point on the parabola is equidistant to the focus and the directrix line. This is a critical property of parabolas and helps understand their geometric structure.
axis of symmetry
The axis of symmetry of a parabola is a vertical line that divides the parabola into two equal mirror-image halves.
For any parabola of the form \( x^{2} = 4py \, \) the axis of symmetry will always be the line \( x = 0 \. \) This line runs vertically through the vertex of the parabola, and in this case, it is the y-axis.
Understanding the axis of symmetry is essential as it helps in graphing the parabola accurately.
It ensures that you know that each side of the parabola is a mirror image of the other. This line also passes through both the vertex and the focus, guiding the symmetry of the whole curve.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Solve each problem. The Roman Colosseum (a) The Roman Colosseum is an ellipse with major axis \(620 \mathrm{ft}\) and minor axis \(513 \mathrm{ft}\). Find the distance between the foci of this ellipse. (b) A formula for the approximate perimeter of an ellipse is $$P=2 \pi \sqrt{\frac{a^{2}+b^{2}}{2}}$$ where \(a\) and \(b\) are the lengths shown in the figure. Use this formula to find the approximate perimeter of the Roman Colosseum.

Write an equation for each parabola with vertex at the origin. through \((\sqrt{3}, 3),\) opening up

Solve each problem. The U.S. Naval Research Laboratory designed a giant radio telescope that had diameter 300 ft and maximum depth 44 ft. (Source: Mar, J., and H. Liebowitz, Structure Technology for Large Radio and Radar Telescope Systems, The MIT Press.) (a) Find the equation of a parabola that models the cross section of the dish if the vertex is placed at the origin and the parabola opens up. (b) The receiver must be placed at the focus of the parabola. How far from the vertex should the receiver be located? (IMAGE CANNOT COPY)

Write an equation for each parabola. vertex \((-5,6),\) directrix \(x=-12\)

Solve each problem. Suppose the two balls are now thrown upward at a \(60^{\circ}\) angle on Mars and the moon. If their initial velocity is \(60 \mathrm{mph},\) then their \(x y\) -coordinates in feet at time \(x\) in seconds can be modeled by the following equation. $$ y=\frac{19}{11} x-\frac{g}{3872} x^{2} $$ (Source: Zeilik, M., and S. Gregory, Introductory Astronomy and Astrophysics, Fourth Edition, Brooks/Cole.) (a) Graph on the same coordinate axes the paths of the balls if \(g=5.2\) for the moon. Use the window \([0,1500]\) by \([0,1000]\) (b) Determine the maximum height of each ball to the nearest foot.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.