/*! 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 54 Classify the graph of the equati... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 54

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$x^{2}+y^{2}-4 x-6 y-23=0$$

Short Answer

Expert verified
The graph of the equation \(x^{2}+y^{2}-4 x-6 y-23=0\) is a circle with center at (2,3) and radius 6.

Step by step solution

01

Completion of Square

Complete the square for the \(x\) and \(y\) terms. Here, the equation can be rewritten as: \((x^{2}-4 x)+(y^{2}-6 y)=-23\). Next, use the formula \((a-b)^{2}=a^{2}-2ab+b^{2}\) to complete the square for \(x\) and \(y\). This implies that \(b\) is half the coefficient of \(x\) and \(y\) respectively. For \(x\), \(b=4/2=2\), and for \(y\), \(b=6/2=3\). The equation now becomes \((x-2)^{2}-4+(y-3)^{2}-9=-23\).
02

Simplify Equation

Combine like terms and simplify the equation to obtain: \((x-2)^{2}+(y-3)^{2}=4+9+23\). Further simplifying it gives: \((x-2)^{2}+(y-3)^{2}=36\).
03

Classify the Graph

The transformed equation \((x-2)^{2}+(y-3)^{2}=36\) is in the form of a circle. In this case, center is at \((h,k)\) = (2,3) and radius \(r\) is \(sqrt{36}=6\).

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

The comet Hale-Bopp has an elliptical orbit with an eccentricity of \(e \approx 0.995 .\) The length of the major axis of the orbit is approximately 500 astronomical units. Find a polar equation for the orbit. How close does the comet come to the sun?

A straight road rises with an inclination of 0.20 radian from the horizontal. Find the slope of the road and the change in elevation over a one-mile stretch of the road.

Find the distance between the point and the line. Point \((1,-3)\) Line \(4 x-3 y=-7\)

In Exercises \(91-116\), convert the polar equation to rectangular form. $$r=\frac{6}{2 \cos \theta-3 \sin \theta}$$

In Exercises \(117-126\), convert the polar equation to rectangular form. Then sketch its graph. $$\theta=3 \pi / 4$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Applied Mathematics

Read Explanation

Probability and 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.