/*! 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 60 Classify the graph of the equati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 60

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$4 x^{2}+25 y^{2}+16 x+250 y+541=0$$

Short Answer

Expert verified
The graph of the given equation is an ellipse

Step by step solution

01

- Rewrite in Standard Form

To rewrite the equation into standard form, complete the square for both \(x\) and \(y\) terms. Begin by grouping the \(x\) and \(y\) terms together: \(4(x^{2} + 4x) + 25(y^{2} + 10y) = -541\). Now complete the square: add \((4/2)^{2} = 4\) to the \(x\) terms and \((10/2)^{2} = 25\) to the \(y\) terms. Remember to also add these terms to the right side. This will result in: \(4(x + 2)^{2} - 16 + 25(y + 2)^{2} - 100 = -541\), which simplifies to: \(4(x + 2)^{2} + 25(y + 2)^{2} = 35\)
02

- Identification

Now, the equation is in standard form where it can be classified. It takes the form of: \(\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1\), which is the standard form of an ellipse, except the right side of the equation equals to 35 rather than 1. To correct this, divide both sides by 35. This process will result in the final standard form of: \(\frac{(x + 2)^{2}}{8.75} + \frac{(y + 2)^{2}}{1.4} = 1\)
03

- Classification

The final standard form equation has a positive \(x^{2}\) term and a positive \(y^{2}\) term, just like the standard form of an ellipse. Also the denominator of \(y^{2}\) is larger than that of \(x^{2}\), which indicates a vertical ellipse. Therefore, the graph of the equation is an ellipse

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 distance between the point and the line. Point \((-1,2)\) Line \(5 x+3 y=-4\)

Find the distance between the point and the line. Point \((1,-3)\) Line \(y=2 x-5\)

In Exercises \(71-90,\) convert the rectangular equation to polar form. Assume \(a > 0\). $$y^{2}=x^{3}$$

Convert the polar equation \(r=\cos \theta+3 \sin \theta\) to rectangular form and identify the graph.

The graph of the parametric equations \(x=t\) and \(y=t^{2}\) is shown below. Determine whether the graph would change for each set of parametric equations. If so, how would it change? (GRAPH CANNOT COPY) (a) \(x=-t, y=t^{2}\) (b) \(x=t+1, y=t^{2}\) (c) \(x=t, y=t^{2}+1\)
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Probability and Statistics

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Statistics

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.