/*! 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 724 Write an equation for the circle... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 42: Problem 724

Write an equation for the circle with center \(\mathrm{A}(-3,-2)\) and radius 3.

Short Answer

Expert verified
\((x + 3)^2 + (y + 2)^2 = 9\)

Step by step solution

01

Identify the center and radius of the circle

The center of the circle, \((x_0, y_0)\), is given as \(\mathrm{A}(-3, -2)\), and the radius, \(r\), is given as 3.
02

Use the standard formula for a circle

The standard formula for a circle is \((x - x_0)^2 + (y - y_0)^2 = r^2\). Plug in the values for the center and radius that we identified in Step 1: \((x + 3)^2 + (y + 2)^2 = 3^2\).
03

Simplify the equation

Simplify the equation: \((x + 3)^2 + (y + 2)^2 = 9\). This is the equation of the circle with center \(\mathrm{A}(-3, -2)\) and radius 3.

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.

Circle Formula
A circle can be represented using an equation in geometry. The standard formula for a circle is \[(x - x_0)^2 + (y - y_0)^2 = r^2\]Here:
  • \( (x_0, y_0) \) represents the coordinates of the center of the circle.
  • \( r \) is the radius of the circle.
  • \( (x, y) \) are the coordinates of any point on the circle.
This formula helps in visualizing a circle on a coordinate plane. By plugging in the center and the radius into the formula, you can easily derive the circle's equation. The equation defines all points \((x, y)\) that lie exactly \( r \)units away from the center \((x_0, y_0)\).Understanding this formula is essential for solving geometry problems involving circles.
Coordinates
In geometry, coordinates are used to specify the location of a point in a plane using an ordered pair. For a circle, the center is expressed as coordinates \((x_0, y_0)\).Coordinates play a crucial role in geometric problem solving. They tell us:
  • The exact location of the center of the circle.
  • Where any point lies in relation to the circle's center.
  • Helps visualize the shape in the coordinate plane.
In our problem,\((-3, -2)\) are the coordinates of the center \(A\).By knowing the coordinates of the center and using the circle formula, you can establish the direction and shape of the circle within the chosen plane.Coordinates are fundamental in forming equations that accurately reflect real-world geometry problems.
Geometry Problem Solving
Geometry problem solving often involves applying formulas and understanding spatial relationships between figures. When tasked to find the equation of a circle:
  • Identify key information: the center and the radius.
  • Use the circle formula to establish the equation.
  • Simplify the equation, making sure each part correctly reflects the circle's properties.
In the given exercise, once coordinates of \((-3, -2)\) and radius \(3\) were identified, the standard circle formula was applied. This provided the circle equation \((x + 3)^2 + (y + 2)^2 = 9\).Solving geometry problems like this one involves careful analysis and a clear understanding of formulas and geometric principles.Practice with various problems helps to strengthen these skills and build confidence in solving complex geometry tasks.

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 area of the ellipses (a) \(\mathrm{x}^{2} / 9+\mathrm{y}^{2} / 25=1\) (b) \(x^{2} / 144+y^{2} / 256=1\) (c) \(x^{2} / 64+y^{2} / 49=1\) (d) \(\mathrm{x}^{2} / 81+\mathrm{y}^{2} / 16=1\)

Find the equation of the ellipse which has vertices \(\mathrm{V}_{1}(-2,6)\), \(\mathrm{V}^{2}(-2,-4)\), and foci \(\mathrm{F}_{1}(-2,4), \mathrm{F}_{2}(-2,-2)\). (See figure,)

The equation of an ellipse is \(\mathrm{x}^{2} / \mathrm{a}^{2}+\mathrm{y}^{2} / \mathrm{b}^{2}=1\). Discuss what happens if \(a=b=r\).

In the equation of an ellipse, \(4 x^{2}+9 y^{2}-16 x+18 y-11=0\), determine the standard form of the equation, and find the values of \(a, b, c\), and \(e\).

Find the points of intersection (if any) of the circles \(\mathrm{C}_{1}\) and \(C_{2}\) where \(C_{1}: x^{2}+y^{2}-4 x-2 y+1=0\) $$ C_{2}: x^{2}+y^{2}-6 x+4 y+4=0 $$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Pure Maths

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.