Problem 41 - Calculus Single Variable

Question

Sketch the given region. \(\left\\{(x, y):(x-2)^{2}+y^{2} \geq 4\right\\}\)

Step-by-Step Solution

Answer

The region includes all points outside and on the circle centered at \((2,0)\) with radius \(2\).

Step 1: Identify the Inequality

The inequality given is \((x-2)^2 + y^2 \geq 4\). This represents a region on the coordinate plane. The expression \((x-2)^2 + y^2\) is similar to the equation of a circle, \((x-h)^2 + (y-k)^2 = r^2\).

Step 2: Determine the Circle's Center and Radius

In the expression \((x-2)^2 + y^2\), compare it with \((x-h)^2 + (y-k)^2 = r^2\). Here, \(h=2\), \(k=0\), and \(r^2 = 4\). Thus, the circle has a center at \((2,0)\) and a radius of \(2\).

Step 3: Describe the Region

The inequality \((x-2)^2+y^2 \geq 4\) describes all points that lie on or outside the circle. The circle itself is defined by \((x-2)^2 + y^2 = 4\). So the region of interest is the area outside and including the boundary of this circle.

Step 4: Sketch the Circle and the Region

Plot the circle with center \((2,0)\) and radius \(2\) on the coordinate plane. The boundary is the circle itself. Shade the region outside the circle to represent the inequality \((x-2)^2 + y^2 \geq 4\), which includes all points not enclosed by the circle.

Key Concepts

Circle Equation

Understanding the equation of a circle is crucial when dealing with problems in coordinate geometry, specifically those involving inequalities in two variables. A standard circle equation is written as

• \((x-h)^2 + (y-k)^2 = r^2\)

This formula represents a circle on an xy-plane with:

• Center at \((h, k)\)
• Radius of \(r\)

The expression

• \((x-2)^2 + y^2\)

is quite similar, where by comparison we identify:

• Center: \((2,0)\)
• Radius: \(2\) since \(r^2 = 4\), making \(r=\sqrt{4} = 2\)

Therefore, the region defined by

• \((x-2)^2 + y^2 \geq 4\)

relates to the properties of this circle. Here, the inequality sign \(\geq\) indicates we're concerned with points lying on or beyond the circle's periphery.

Coordinate Geometry

Coordinate geometry involves plotting and analyzing points, lines, and shapes on a coordinate plane. Key features of coordinate geometry include:

• Two-dimensional space divided into \(x\) (horizontal) and \(y\) (vertical) axes.
• Each point represented as an ordered pair \((x, y)\).
• Understanding graphical representation of algebraic equations.

For equations of a circle, it becomes an essential tool to determine locations and regions as described by the equations. Specifically, for

• \((x-2)^2 + y^2 \geq 4\)

this translates to finding points that satisfy these conditions within the coordinate plane. By identifying the center as \((2,0)\), any point \((x,y)\) is evaluated based on its distance from the center, ensuring it meets the condition set by the inequality. This demonstrates the power of coordinate geometry as a visual and analytical method for understanding how equations translate to physical regions on the plane.

Sketching Regions

Sketching regions is a powerful skill to visually understand inequalities and the areas they cover on a graph. To start sketching the region defined by the inequality

• \((x-2)^2 + y^2 \geq 4\)

follow these steps:

• First, draw the circle described by the equation \((x-2)^2 + y^2 = 4\). This involves plotting the center \((2,0)\) and marking all points a radius of \(2\) from it, which forms the circle's boundary.
• Since the inequality states \(\geq\), shade the area outside this circle, indicating all regions where the expression holds true.

The point on or outside the circle represent the solutions to the inequality. The boundary circle itself is also part of the solution because of the inclusive inequality sign. By accurately drawing, these visualizations help to solidify the concepts of spatial areas determined by inequalities and enhance overall comprehension of how algebraic conditions impact graphical shapes.