/*! 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 35 Find the standard form of the eq... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 35

Find the standard form of the equation of the hyperbola with the given characteristics. Vertices: (2,0),(6,0)\(;\) foci: (0,0),(8,0)

Short Answer

Expert verified
The standard form of the equation of the hyperbola is (x - 4)^2/4 - y^2/12 = 1.

Step by step solution

01

Calculate the center of the hyperbola

The center of the hyperbola is the midpoint of the line segment connecting the vertices. As the vertices given are (2,0) and (6,0), the center of the hyperbola is \(\left(\frac{2+6}{2} , \frac{0+0}{2}\right)\), which equals (4,0)
02

Find the value of a

The distance from the center of the hyperbola to each vertex is 'a'. This distance can be calculated using the distance formula or by simple subtraction since the center and vertices lie along the same line. The point (4,0) is 2 units from (2,0) and 2 units from (6,0). So, 'a' = 2.
03

Find the value of c

The foci are at a distance 'c' from the center along the major axis. The given foci are at (0,0) and (8,0). The distance from the center (4,0) to each foci is 4 units, so 'c' = 4.
04

Find the value of b

For a hyperbola, the relationship between 'a', 'b', and 'c' is given by the equation c^2 = a^2 + b^2. Solving for 'b' gives us: b^2 = c^2 - a^2. Substituting the given values we have: b^2 = 4^2 - 2^2, which simplifies to b^2 = 12.
05

Write down the standard form of the equation

The standard form of a hyperbola's equation with its center at (h,k) is (x - h)^2/a^2 - (y - k)^2/b^2 = 1. Substituting the values that we calculated, we find: (x - 4)^2/2^2 - (y - 0)^2/12 = 1 . This simplifies to: (x - 4)^2/4 - y^2/12 = 1.

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

Find the vertex, focus, and directrix of the parabola, and sketch its graph. \(x=\frac{1}{4}\left(y^{2}+2 y+33\right)\)

Find an equation of the tangent line to the parabola at the given point, and find the \(x\) -intercept of the line. \(y=-2 x^{2},(2,-8)\)

Find the vertex, focus, and directrix of the parabola, and sketch its graph. \(y^{2}+6 y+8 x+25=0\)

Determine whether the statement is true or false. Justify your answer. It is possible for a parabola to intersect its directrix.

The revenue \(R\) (in dollars) generated by the sale of \(x\) units of a digital camera is given by \((x-135)^{2}=-\frac{5}{7}(R-25,515)\) Use a graphing utility to graph the function and approximate the number of sales that will maximize revenue.
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

Read Explanation

Logic and Functions

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.