/*! 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 129 Graph the equation. $$ (x+5)... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 129

Graph the equation. $$ (x+5)^{2}+(y+2)^{2}=4 $$

Short Answer

Expert verified
The circle's center is at \( (-5, -2) \) with a radius of 2 units.

Step by step solution

01

Recognize the Equation Type

Identify that the equation \[ (x+5)^{2}+(y+2)^{2}=4 \] represents a circle. The standard form of a circle's equation is \[ (x-h)^{2}+(y-k)^{2}=r^2, \] where \( (h, k) \) is the center and \( r \) is the radius.
02

Determine the Center

Compare the given equation to the standard form. From \[ (x+5)^{2}+(y+2)^{2}=4, \] recognize that \( h = -5 \) and \( k = -2. \) Hence, the center of the circle is \( (-5, -2). \)
03

Find the Radius

Identify the radius by comparing the given equation to the standard form. The term on the right side of the equation \[ r^2 = 4 \] allows you to find the radius as \( r = \sqrt{4} = 2. \)
04

Plot the Center

On a coordinate plane, plot the center of the circle at the point \( (-5, -2). \)
05

Draw the Circle

With the center plotted, use the radius of 2 units to draw a circle around the center. Ensure that the distance from the center to any point on the circle is exactly 2 units.

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Key Concepts

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

Standard Form of a Circle
The standard form of a circle's equation is essential for recognizing and graphing circles in coordinate geometry. The form is \((x-h)^{2}+(y-k)^{2}=r^2\), where \( (h, k) \) represents the coordinates of the circle's center, and \( r \) stands for the radius. This form helps easily identify the circle's critical components. By comparing a given equation, like \((x+5)^{2}+(y+2)^{2}=4\), to the standard form, you can quickly determine the center and radius. Remember, \( h \) and \( k \) are the coordinates, and \( r \) is derived from the square root of the number on the right side.
Coordinates
Coordinates are crucial in geometry for defining locations on a plane. In the context of a circle's equation, the coordinates \( (h, k) \) specify the circle's center. For example, in the equation \((x+5)^{2}+(y+2)^{2}=4\), comparing to the standard form tells us that \((h, k) = (-5, -2)\). This means the center of the circle is at the point \( (-5, -2) \). When plotting this on a coordinate plane, you start with these coordinates as your reference point. Use this center to measure the radius and draw the circle accurately by ensuring every point on the circle is equidistant from this center.
Radius and Center
Understanding the radius and center is vital for accurately graphing a circle. The radius is the distance from the center to any point on the circle. For the given equation \((x+5)^{2}+(y+2)^{2}=4\), we find that \( r = \sqrt{4} = 2\). This tells us the circle has a radius of 2 units. The center, as determined earlier, is \( (-5, -2) \). To draw the circle:
  • Start by plotting the center at \( (-5, -2) \).
  • Then, from this center, use a compass or measure 2 units in all directions to mark the edge of the circle.
These components allow you to visualize and graph the circle precisely as a round shape with the specified radius and center coordinates.

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Most popular questions from this chapter

(a) write the equation in standard form and (b) graph. $$ 9 x^{2}-4 y^{2}-18 x+8 y-31=0 $$

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