/*! 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 52 Sketch the graph of each equatio... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 52

Sketch the graph of each equation. If the graph is a parabola, find irs vertex. If the graph is a circle, find its center and radius. $$x^{2}+y^{2}+6 x+10 y-2=0$$

Short Answer

Expert verified
Center: (-3, -5), Radius: 6.

Step by step solution

01

Identify the Equation Type

The given equation is \(x^2 + y^2 + 6x + 10y - 2 = 0\). It contains both \(x^2\) and \(y^2\) terms with equal coefficients, indicating that it's a circle.
02

Rearrange and Group Terms

To rearrange and group the quadratic terms, express the equation in a way that allows for completing the square:\[ x^2 + 6x + y^2 + 10y = 2 \]
03

Complete the Square for x-terms

For the \(x\)-terms, we have \(x^2 + 6x\). To complete the square, take half of the coefficient of \(x\) (which is 3), square it, and add and subtract it inside the expression: \(x^2 + 6x = (x+3)^2 - 9\).
04

Complete the Square for y-terms

For the \(y\)-terms, we have \(y^2 + 10y\). Take half of the coefficient of \(y\) (which is 5), square it, and add and subtract it:\(y^2 + 10y = (y+5)^2 - 25\).
05

Combine and Simplify the Equation

Substitute the completed squares back into the equation:\[(x + 3)^2 - 9 + (y + 5)^2 - 25 = 2 \]Combine and simplify:\[(x + 3)^2 + (y + 5)^2 = 36\]
06

Identify the Center and Radius

The equation \((x + 3)^2 + (y + 5)^2 = 36\) represents a circle with center at \((-3, -5)\) and radius \(\sqrt{36} = 6\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Completing the Square
Completing the square is a mathematical method used to transform a quadratic expression into a perfect square trinomial, making it easier to analyze or graph. The main idea is to restructure the quadratic terms in an equation so that they form a square binomial.

Here's how it works:
  • Consider a generic quadratic expression in terms of one variable, such as \(x^2 + bx\).
  • Take half of the coefficient of \(x\), which is \(\frac{b}{2}\), and square it to complete the square.
  • Add and subtract this squared value in the expression to form a perfect square trinomial. This can be expressed as \((x + \frac{b}{2})^2 - (\frac{b}{2})^2\).
By completing the square for both \(x\) and \(y\) terms in the circle equation \(x^2 + y^2 + 6x + 10y - 2 = 0\), we can rewrite it in the form \((x + 3)^2 + (y + 5)^2 = 36\), which reveals key elements of the graph.
Circle Graphing
Graphing a circle involves plotting all the points in the plane that are equidistant from a central point, known as the center. The equation of a circle in standard form is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.

To graph the circle from the transformed equation \((x + 3)^2 + (y + 5)^2 = 36\):
  • First, recognize that the circle's center is at \((-3, -5)\) because the equation is in the form \((x-h)^2 + (y-k)^2 = r^2\) and looks like \((x - (-3))^2 + (y - (-5))^2\).
  • The radius is the square root of 36, which is 6.
  • Once you have the center and radius, plot the center on the coordinate plane, and draw a circle around it with the specified radius.
Center and Radius
The center and radius of a circle provide a lot of information that makes graphing straightforward and provide insights into the geometry of the circle.

Let's break down why the center and radius are so crucial:
  • The center, given by \( (h, k) \), is a fixed point from which all points on the circle are equidistant.
  • The radius, represented by \( r \), is that consistent distance from the center to any point on the circle. It's always positive as a measure of length.
From the final step of our solution, the circle equation is \((x + 3)^2 + (y + 5)^2 = 36\). Thus, the center is \((-3, -5)\), and the radius is determined through \(\sqrt{36}\), resulting in a radius of 6.

Understanding these aspects enables one to construct the circle accurately and appreciate its geometric properties.

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

$$ \text { Graph the system }\left\\{\begin{array}{l} {y=x^{2}} \\ {y \geq x+2} \\ {x \geq 0} \\ {y \geq 0} \end{array}\right. $$

Sketch the graph of each equation. If the graph is a parabola, find irs vertex. If the graph is a circle, find its center and radius. $$5 x^{2}+5 y^{2}=25$$

Perform the indicated operations. \(2 x^{3}-4 x^{3}\)

Graph each equation. See Sections 3.2 and 3.3 $$y=-3 x+3$$

The graph of each equation is a parabola. Find the vertex of the parabola and sketch its graph. See Examples I through 4. $$ x=(y-2)^{2}+3 $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Pure Maths

Read Explanation

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

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.