/*! 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 31 For Problems \(21-32\) a. Draw... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 31

For Problems \(21-32\) a. Draw a sketch showing the given information. Sketch the conic section. b. Find the particular equation (Cartesian or parametric). c. Plot the graph on your grapher. Does your sketch in part a agree with the plot? Parabola with focus (2,3) and directrix \(y=5\)

Short Answer

Expert verified
Sketch a downward opening parabola with focus (2,3) and directrix y=5. The vertex is at (2,4), and the particular equation of the parabola is \(y-4)^2 = -4(x-2)\). The graph of this equation should agree with the initial sketch.

Step by step solution

01

Understanding the Conic Section

Identify the conic section as a parabola. Recognize that the focus is a point where all points on the parabola are equidistant from the directrix, a line. The given focus is (2,3) and the directrix is the line y=5.
02

Sketching the Parabola

Sketch the directrix, a horizontal line at y=5, and the focus at (2,3). Sketch the parabola such that it opens downwards (since the focus is below the directrix) and ensure it is equidistant at all points from the focus and directrix.
03

Determining the Vertex

The vertex lies midway between the focus and the directrix. Since the focus is at (2,3) and the directrix is at y=5, the y-coordinate of the vertex must be the average of 3 and 5 which is 4. The x-coordinate remains the same as the focus's so the vertex is at (2,4).
04

Finding the Equation of the Parabola

Use the standard form of the equation of a parabola that opens downward: \(y-k)^2 = -4p(x-h)\), where (h,k) is the vertex and p is the distance from the vertex to the focus. Substitute the vertex (2,4) into the equation: \(y-4)^2 = -4p(x-2)\). The distance from the vertex to the focus is 1 (since the vertex is at y=4 and focus at y=3), so p=1 and the equation becomes \(y-4)^2 = -4(x-2)\).
05

Graphing the Parabola

Using a graphing software or graphing calculator, plot the parabola equation \(y-4)^2 = -4(x-2)\). Adjust the viewing window to include the vertex, focus, and a sufficient part of the parabola to see its shape.
06

Comparing Sketch and Graph

Compare the sketched parabola from Step 2 with the plotted graph from Step 5. Look for similarities in the shape, orientation, location of vertex, focus and directrix. The sketch should match the plotted graph if both are constructed correctly.

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.

Conic Sections
Conic sections are the curves obtained by intersecting a cone with a plane. They include circles, ellipses, parabolas, and hyperbolas, depending on the angle at which the plane cuts the cone.

When the plane intersects parallel to the slope of the cone, a parabola is formed. A parabola's distinct U-shape is symmetrical, and it can open upwards, downwards, or sideways depending on its orientation. This property makes them applicable in various real-world contexts, such as satellite dishes and car headlights, where the focus of a parabola plays a crucial role in directing signals and light.
Focus and Directrix of a Parabola
Every parabola has a focus and a directrix which define its precise shape. The focus is a fixed point inside the parabola, and the directrix is a line outside it. The distance of any point on the parabola to the focus is equal to the perpendicular distance to the directrix. This equivalence maintains the parabola's curved shape.

For example, a parabola with a focus at (2,3) and a directrix at \(y=5\) would curve downwards, as the focus lies below the directrix. It is essential to understand the spatial relationship between the focus and directrix to graph parabolas accurately, as in the given exercise.
Standard Form of a Parabola Equation
The standard form of a parabola's equation is commonly expressed as \(y-k)^2=4p(x-h)\) or \(x-h)^2=4p(y-k)\), where \(h,k\) is the vertex of the parabola and \(p\) represents the distance from the vertex to the focus. The sign and coefficient of the term \(4p\) determine the direction the parabola opens.

An equation like \(y-4)^2=-4(x-2)\) indicates a parabola that opens downwards, with the vertex at (2,4). By mastering this standard form, one can graph a parabola by just knowing the vertex and the distance to the focus or directrix.
Vertex of a Parabola
The vertex of a parabola is the peak or the point where the curve turns; it is also the highest or lowest point depending on the orientation of the parabola. For parabolas that open upwards or downwards, the vertex lies on the axis of symmetry.

In the example provided, the vertex is calculated as the average of the focus and directrix on the y-axis, with the x-coordinate shared with the focus. The vertex at (2,4), reveals valuable information as it not only provides a starting point for graphing but also determines the parabola's axis of symmetry and the direction in which it opens.

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

Conic Construction Problem 1: Plot on graph paper the conic with focus \((0,0),\) directrix \(x=-6,\) and eccentricity \(e=2 .\) Put the \(x\) -axis near the middle of the graph paper and the \(y\) -axis just far enough from the left side to fit the directrix on the paper. Plot the points for which \(d_{1}\) from the directrix equals 2,4,6,8 and \(10 .\) Connect the points with a smooth curve. Which conic section have you graphed?

Use the discriminant to determine which conic section the graph will be. Confirm your conclusion by plotting the graph Sketch the result. (In your sketch, connect any gaps left by low-resolution graphers.) $$3 x^{2}-10 x y+6 y^{2}-12 x+4 y+10=0$$

Use the discriminant to determine which conic section the graph will be. Confirm your conclusion by plotting the graph Sketch the result. (In your sketch, connect any gaps left by low-resolution graphers.) $$x^{2}+6 x y+9 y^{2}-3 x-4 y-10=0$$

Mars Orbit Problem: Mars is in an elliptical orbit around the Sun, with the Sun at one focus. The aphelion (the point farthest from the Sun) and the perihelion (the point closest to the Sun) are 155 million miles and 128 million miles, respectively, as shown in 1\. 41 (not to scale) a. How long is the major axis of the ellipse? What is the major radius? b. Find the focal radius and the minor radius of the ellipse. c. Write a Cartesian equation for the ellipse, with the center at the origin and the major axis along the \(x\) -axis. d. At the two equinoxes (times of equal day and night), the angle at the Sun between the major axis and Mars is \(90^{\circ} .\) At these times, what is the value of \(x ?\) How far is Mars from the Sun? e. Find the eccentricity of the ellipse. f. How far from the Sun is the closer directrix of the ellipse? g. Write parametric equations for the ellipse. Plot the graph using parametric mode. Zoom appropriately to make equal scales on the two axes. h. The ellipse you plotted in part g looks almost circular. How do the major and minor radii confirm this? How does the eccentricity confirm this?

a. Name the conic section simply by looking at the Cartesian equation. b. Sketch the graph. c. Transform the given equation to an equation of the form $$ A x^{2}+B x y+C y^{2}+D x+E y+F=0 $$ d. Plot the Cartesian equation using the result of part c. Does it agree with part b? $$0.2(x-1)^{2}+(y-6)=0$$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Applied Mathematics

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.