Problem 25 - A Graphical Approach to College Algebra

Question

Use the concepts of this section to work. 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

Step-by-Step Solution

Answer

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

Step 1: Understand the Standard Form of a Circle

The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is \((x-h)^2 + (y-k)^2 = r^2\). The inequality \((x-h)^2 + (y-k)^2 < r^2\) describes the region inside this circle.

Step 2: Identify the Center and Radius from the Inequality

Given the inequality \((x-5)^2 + (y-2)^2 < 4\), we can compare it to the standard form of a circle \((x-h)^2 + (y-k)^2 < r^2\). Here, \(h = 5\), \(k = 2\), and \(r^2 = 4\). Thus, \(r = \sqrt{4} = 2\).

Step 3: Interpret the Circle Description from the Inequality

The inequality \((x-5)^2 + (y-2)^2 < 4\) represents all the points that are inside a circle with a center \((5, 2)\) and a radius of 2. It describes the interior of the circle excluding the boundary.

Step 4: Choose the Correct Answer Option

Based on the interpretation in Step 3, the circle has a center at \((5, 2)\) and a radius of 2. Thus, the correct description of the inequality is option B: "The region inside a circle with center \((5, 2)\) and radius 2."

Key Concepts

Circle Equation

A circle equation in its standard form is a powerful way to define the locus of points that are equidistant from a fixed point, known as the center. This equation is represented by the formula \[(x-h)^2 + (y-k)^2 = r^2\]where

• \( (h, k) \) is the center of the circle, and
• \( r \) is the circle's radius.

The equation captures all points \((x, y)\) that lie exactly on the circle. If you modify this equation to an inequality, it broadens the description to include not just the circle itself but potentially its interior or exterior as well.

Inequality Description

When we talk about inequalities of the form \[(x-h)^2 + (y-k)^2 < r^2\]it describes a region, specifically, the area inside the circle. Unlike a strict equation that signifies only the boundary, an inequality like this one includes all the points within that distance, making it inclusive of the circle's interior.

Conversely, if it were \[(x-h)^2 + (y-k)^2 > r^2\]this would represent the region outside the circle, where points are more distant from the center than the radius.

• The inequality symbol \("<"\) signifies points inside the circumference.
• The inequality symbol \(">"\) includes points outside the circumference.

Using inequalities, one can effectively describe various spatial relations in coordinate geometry.

Center and Radius Identification

Identifying the center and radius from the circle inequality is straightforward when you match it to the standard form. Let's take \[(x-5)^2 + (y-2)^2 < 4\].By comparing it to the template \[(x-h)^2 + (y-k)^2 = r^2\],we find:

• \( h = 5 \), indicating the x-coordinate of the center.
• \( k = 2 \), indicating the y-coordinate of the center.
• \( r^2 = 4 \) gives the radius as \( r = \sqrt{4} = 2 \).

With these values, you can quickly determine that the circle's center is located at \((5, 2)\) with a radius of 2 units. This makes interpreting the inequality easier as you proceed to plot or describe it geometrically.

Graph Interpretation

Graphing inequalities involving circles helps in visualizing the set of points described. For \((x-5)^2 + (y-2)^2 < 4\):

• You would plot the center at \((5, 2)\).
• Then, draw a circle with a radius of 2.
• Since the inequality symbol is \("<"\), you shade the region inside this circle.

This graph illustrates all points within 2 units from the center. The boundary is not included since the inequality is strict, not allowing equality ("=") at the circle's edge. Graphing it this way provides not only mathematical correctness but also a visual guide, making it easier to comprehend spatial relations between points and shapes in the coordinate plane.