/*! 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 71 Write the standard form of the e... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 71

Write the standard form of the equation of a parabola whose points are equidistant from \(y=4\) and \((-1,0)\)

Short Answer

Expert verified
The standard form of the equation of the parabola is \((x+1)^2 = 8(y-2)\)

Step by step solution

01

Identify the focus and the directrix

The focus of the parabola is given as \((-1,0)\) and the directrix is the line \(y=4\), which is a horizontal line.
02

Calculate the vertex of the parabola

The vertex of a parabola is located midway between the focus and the directrix. As the focus is \((-1,0)\) and the directrix is \(y=4\), the vertex is halfway between \(y=0\) (the y-coordinate of the focus) and \(y=4\). Therefore, the y-coordinate of the vertex is \((0+4)/2 = 2\). The x-coordinate of the vertex is always the same as the x-coordinate of the focus, which is \(-1\). Thus, the vertex is \((-1,2)\).
03

Use the vertex and focus to calculate the value of p

The value of \(p\) is the distance from the vertex to the focus or to the directrix. Hence, \(p = 2-0 = 2\).
04

Write the equation of the parabola in standard form

Since the vertex of the parabola is \((-1,2)\), and the parabola opens downwards (since the focus is below the directrix), the equation of the parabola is written in the form \((x-h)^2 = 4p(y-k)\). Plugging in the values found gives \((x+1)^2 = 4*2(y-2)\). Simplifying this gives the final equation \((x+1)^2 = 8(y-2)\).

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!

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

Use the center, vertices, and asymptotes to graph each hyperbola. Locate the foci and find the equations of the asymptotes. $$(x-3)^{2}-4(y+3)^{2}=4$$

Graph each ellipse and give the location of its foci. $$\frac{(x-1)^{2}}{16}+\frac{(y+2)^{2}}{9}=1$$

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the hyperbola. Locate the foci and find the equations of the asymptotes. $$16 x^{2}-y^{2}+64 x-2 y+67=0$$

Convert each equation to standard form by completing the square on \(x\) and \(y .\) Then graph the ellipse and give the location of its foci. $$9 x^{2}+16 y^{2}-18 x+64 y-71=0$$

An explosion is recorded by two microphones that are 1 mile apart. Microphone \(M_{1}\) received the sound 2 seconds before microphone \(M_{2} .\) Assuming sound travels at 1100 feet per second, determine the possible locations of the explosion relative to the location of the microphones.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Calculus

Read Explanation

Theoretical and Mathematical Physics

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.