Problem 37 - A Graphical Approach to Precalculus with Limits

Question

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: Identify the Inequality's Form

The inequality \((x-5)^2 + (y-2)^2 < 4\) is similar to the standard form of a circle's equation, which is \((x-h)^2 + (y-k)^2 = r^2\). In this case, \(r^2\) is replaced by an inequality to indicate the region inside the circle.

Step 2: Determine the Circle's Center

Compare \((x-5)^2 + (y-2)^2 < 4\) to the standard circle equation \((x-h)^2 + (y-k)^2 = r^2\). From this, \(h = 5\) and \(k = 2\). Therefore, the center of the circle is \((5, 2)\).

Step 3: Find the Radius

The inequality \((x-5)^2 + (y-2)^2 < 4\) can be converted into the equation \(r^2 = 4\) to determine the radius. Thus, \(r = \sqrt{4} = 2\).

Step 4: Describe the Region

Since the inequality is \(<\), it describes the region inside the circle with center \((5, 2)\) and radius 2.

Step 5: Select the Correct Answer

From the options given, B is the correct description: "The region inside a circle with center \((5, 2)\) and radius 2."

Key Concepts

Circle Equations

When we discuss circle equations in coordinate geometry, we delve into a fascinating aspect of mathematics that involves shapes and their properties on a plane. The general equation of a circle is given by \[(x-h)^2 + (y-k)^2 = r^2\] where:

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

In this form, each point \((x, y)\) lies on the circumference of the circle. The distance from any point \((x, y)\) on the circle to the center \((h, k)\) is always equal to the radius. This provides a glimpse into the structured beauty of circle equations. They graph as perfect circles on a coordinate plane, demonstrating the simplicity and elegance of geometry.

Inequalities

Inequalities are essential in the study of mathematical expressions and equations. They provide ways to express the vast range of values that satisfy certain conditions, as opposed to equalities, which pinpoint exactness. In the context of graphing, inequalities permit us to depict regions rather than specific points. When we integrate inequalities into circle equations like \[(x-5)^2 + (y-2)^2 < 4\]we define a region inside the circle. This is because any point \((x, y)\) satisfying this inequality is less than the radius squared distance from the center \((5, 2)\). Thus, such inequalities help in understanding not just the boundary of the geometric figure but the area within. Think of it as shading a region in a graph where the inequality solutions reside, emphasizing the vastness that inequalities can cover.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the synergy of algebra and geometry. This branch of mathematics forms a bridge allowing the application of algebraic equations to geometric interpretations. Through this, we gain tools like the distance formula, the midpoint formula, and the slope concept to analyze shapes and their properties on graphs.In coordinate geometry, the equation of a circle not only identifies the circle but also locates it on a coordinate plane. By using the equation \[(x-5)^2 + (y-2)^2 = r^2\]we clearly visualize a circle centered at \((5, 2)\) with a radius \(r\). Each modification to \(h\), \(k\), or \(r\) shifts the circle's position or changes its size. Thus, this mathematical playground empowers us to describe and solve problems involving curves, distances, and shapes more effectively. Understanding coordinate geometry opens a window to visualizing algebraic equations and provides clarity to graphical problems.