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Chapter 1: Problem 60

Find the center and the radius of the circle given by the equation \((x+1)^{2}+(y-3)^{2}=9\).

Short Answer

Expert verified
The center of the circle is \\( (-1, 3) \\\) and the radius is 3.

Step by step solution

01

Identify the Standard Form of the Circle Equation

The given equation of the circle is \( (x + 1)^{2} + (y - 3)^{2} = 9 \). This is already in the standard form for a circle \( (x - h)^{2} + (y - k)^{2} = r^{2} \), where \( (h, k) \) is the center and \( r \) is the radius.
02

Compare and Extract the Center (h, k)

By comparing the given equation \( (x + 1)^{2} + (y - 3)^{2} = 9 \) with the standard form \( (x - h)^{2} + (y - k)^{2} = r^{2} \), we recognize that \( x + 1 \) equates to \( x - (-1) \), giving \( h = -1 \), and \( y - 3 \) indicates \( k = 3 \). Thus, the center of the circle is \( (-1, 3) \).
03

Calculate the Radius

From the standard form \( r^{2} = 9 \), we find that the radius \( r = \sqrt{9} \), which is \( r = 3 \). Thus, the radius of the circle is 3.

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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 Equation
The standard form of a circle's equation helps us easily identify the essential properties of the circle, such as the center and radius. A circle's equation in standard form is expressed as
  • \[(x - h)^{2} + (y - k)^{2} = r^{2}\]
  • Here,
    • \((h, k)\) represents the center of the circle.
    • \(r\) is the radius.
This form allows us to "see" the circle's properties at a glance by just comparing it with a given equation in a similar structure.

For example, the given exercise equation \[(x + 1)^{2} + (y - 3)^{2} = 9\]is already in the standard form. By comparing each part, we easily identify the values
  • \(h = -1\)
  • \(k = 3\)
  • \(r^{2} = 9\)
This knowledge sets the stage for determining the circle's center and radius.
Center of a Circle
The center of a circle is a critical point that defines its position on a coordinate plane. It is represented by the coordinates \((h, k)\) in the circle's standard equation.

From our exercise, if the equation is\[(x + 1)^{2} + (y - 3)^{2} = 9\],compare it with \[(x - h)^{2} + (y - k)^{2} = r^{2}\].You observe that:
  • \((x + 1)^{2}\) implies \(h = -1\) as it is equivalent to \((x - (-1))^{2}\).
  • \((y - 3)^{2}\) indicates \(k = 3\).
Therefore, the center of this circle is \((-1, 3)\).

Understanding how to identify the center of a circle from its equation enhances your ability to manipulate and graph circles, providing insights into their geometric relationships on a plane.
Radius of a Circle
The radius of a circle is a line segment that connects the circle's center to any point on its circumference. In the circle's standard form equation, the radius is represented as \(r\), and the equation provides \(r^{2}\).

Using the exercise's equation:\[(x + 1)^{2} + (y - 3)^{2} = 9\],the part that describes the radius is\(r^{2} = 9\).To find \(r\), solve for the square root of 9:
  • \(r = \sqrt{9} = 3\)
The radius is 3 units.

Knowing the radius allows you to determine the circle's scale and size, making it easier to visualize and graph. It's a simple yet powerful piece of information that completes understanding any circle's geometric shape.

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Most popular questions from this chapter

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sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=-\exp (x) $$

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