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Chapter 9: Problem 23

Work each problem. Which one of the following is a description of the graph of the inequality $$(x-5)^{2}+(y-2)^{2}<4 ?$$ A. the region inside a circle with center \((-5,-2)\) and radius 2 B. the region inside a circle with center \((5,2)\) and radius 2 C. the region inside a circle with center \((-5,-2)\) and radius 4 D. the region outside a circle with center \((5,2)\) and radius 4

Short Answer

Expert verified
B. The region inside a circle with center (5,2) and radius 2

Step by step solution

01

Identify the Standard Form of the Circle Equation

Recall that the standard form for the equation of a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\).
02

Compare Given Inequality to Standard Form

The given inequality is \((x - 5)^2 + (y - 2)^2 < 4\). Identify and compare the components. Here, \(h = 5\), \(k = 2\), and the radius squared \(r^2 = 4\). Thus, \(r = \sqrt{4} = 2\).
03

Determine the Center and Radius

We have the center of the circle at \(5, 2\) and the radius is 2.
04

Interpret the Inequality

The inequality \((x - 5)^2 + (y - 2)^2 < 4\) represents the region inside the circle because the inequality is 'less than'.
05

Choose the Correct Option

Among the given options, B correctly describes the inequality: the region inside a circle with center \(5, 2\) and radius 2.

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Key Concepts

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

circle equation
A circle's equation in the standard form is key to understanding its geometry. It is generally written as \((x - h)^2 + (y - k)^2 = r^2\). Here, \((h, k)\) represents the center of the circle, while \(r\) denotes the radius.
This equation tells us that every point \((x, y)\) on the circle is at a distance \(r\) from the center \((h, k)\). By rearranging this formula, you can easily find the center and radius if the equation is provided in a different form.
For inequalities like \((x - 5)^2 + (y - 2)^2 < 4\), the expression changes from equality \( = \) to an inequality \( < \) or \(\textgreater \), representing regions inside or outside the circle.
inequality solutions
Inequalities in circle equations represent different regions relative to the circle. When the inequality is '<', it means that the solution includes all the points inside the circle, not just on the perimeter.
For example, \((x - 5)^2 + (y - 2)^2 < 4\) tells us that the solution set consists of all points \((x, y)\) that are less than 2 units from the center \((5, 2)\). This means any point within this circle satisfies the inequality.
On the other hand, if the inequality were '>', it would signify all points outside the circle but not on its edge. Understanding these simple notations is essential for accurately graphing and interpreting inequalities involving circles.
radius and center of a circle
To solve and understand problems involving circle inequalities, you must identify the center and radius from the given equation. This lesson focuses on breaking down \((x - h)^2 + (y - k)^2 = r^2\):
  • The center of the circle is \((h, k)\). For instance, in the inequality \((x - 5)^2 + (y - 2)^2 < 4\), the center is \((5, 2)\).
  • The radius \(r\) is found by taking the square root of \(r^2\). In our example, \(\begin{align*}r^2 = 4 \ \rightarrow r = \sqrt{4} = 2\). This gives us a radius of 2 units.

Once you have these two pieces of information, you can easily describe or graph the circle. The center tells you where the circle is positioned, and the radius determines its size.

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