Problem 89 - Algebra and Trigonometry

Question

\(87-90\) . Sketch the region given by the set. $$ \left\\{(x, y) | 1 \leq x^{2}+y^{2}<9\right\\} $$

Step-by-Step Solution

Answer

Draw a shaded annular region between circles of radius 1 and 3, including the inner circle's boundary and excluding the outer circle's boundary.

Step 1: Understand the Inequality

The expression \(1 \leq x^2 + y^2 < 9\) represents a range of values for \(x^2 + y^2\), which is used to describe a region on the coordinate plane. The inequality suggests that \(x\) and \(y\) form a set of points that are at least 1 unit away from the origin but less than 3 units away from the origin, creating a ring or annulus.

Step 2: Identify Boundaries of the Region

The inequality can be split into two parts: \(x^2 + y^2 \geq 1\) and \(x^2 + y^2 < 9\). The equation \(x^2 + y^2 = 1\) represents a circle centered at the origin with radius 1. The equation \(x^2 + y^2 = 9\) represents a circle with radius 3. Therefore, the annular region lies between these two circles.

Step 3: Sketch the Inner Circle

Draw the circle \(x^2 + y^2 = 1\) on the coordinate plane. This is a solid line because the inequality includes \(1\) using \(\leq\), indicating that points on this circle are part of the solution set.

Step 4: Sketch the Outer Boundary

Next, draw the circle \(x^2 + y^2 = 9\) on the coordinate plane. This will be a dashed line since the inequality is \(< 9\) indicating points on this circle are not included in the solution set.

Step 5: Shade the Annular Region

Shade the region between the two circles you have drawn. This represents all the points \((x,y)\) such that \(1 \leq x^2 + y^2 < 9\). This shaded region is the annulus or ring-shaped area.

Key Concepts

Coordinate Plane

The coordinate plane is a fundamental concept in plane geometry that provides a systematic way of describing locations in two-dimensional space. It consists of two perpendicular axes: the horizontal axis called the x-axis, and the vertical axis called the y-axis. These axes intersect at a point known as the origin, denoted as (0,0).
When dealing with exercises involving geometry, the coordinate plane allows us to plot points defined by pairs of numerical coordinates (x, y). Each of these coordinates represents a specific distance from the origin along the respective axis.

• The x-coordinate indicates the point's horizontal position.
• The y-coordinate indicates the point's vertical position.

In problems like the one presented, you're often asked to sketch regions based on inequalities. Here, we make use of the coordinate plane to visually represent and solve such problems. It transforms abstract equations into tangible shapes and areas that can be analyzed further. For instance, circles and other geometric shapes can easily be plotted using these coordinates, enabling deeper exploration and understanding of geometric concepts.

Inequalities in Geometry

Inequalities in geometry describe a range of potential values that certain geometric figures or points can take on the coordinate plane. Instead of just representing a single line or boundary, inequalities allow us to define entire areas or regions. These can include:

• Regions inside or outside of a circle.
• Space between two curves or shapes.

In the exercise, the inequality given is \(1 \leq x^2 + y^2 < 9\). This dictates that the region of interest includes all points where the sum of the squares of x and y is at least 1 but less than 9.
To interpret this:

• The inequality \(x^2 + y^2 \geq 1\) represents all points at least 1 unit away from the origin.
• Meanwhile, \(x^2 + y^2 < 9\) indicates points less than 3 units away from the origin.

The result is an annular region, or ring, lying between two circles — one with radius 1 and the other with radius 3. Inequalities are powerful tools in geometry, allowing us to limit regions to specific areas, creating solutions that fulfill geometric conditions.

Circle Equations

Understanding circle equations is crucial when working with geometric problems involving areas or regions on the coordinate plane. The standard form for a circle centered at the origin (0,0) is given by the equation: \(x^2 + y^2 = r^2\), where \(r\) is the radius of the circle. This simple equation describes all points (x, y) that lie exactly \(r\) units away from the origin.
In more complex problems, circle equations can also be used in inequalities to express regions inside or outside a circle:

• \(x^2 + y^2 \leq r^2\) describes all points inside or on the circle.
• \(x^2 + y^2 < r^2\) describes all points strictly inside the circle but not on the boundary.

In the exercise, two circle equations were essential:

• \(x^2 + y^2 = 1\) defines the boundary of the smaller circle centered at the origin with a radius of 1.
• \(x^2 + y^2 = 9\) defines the boundary of the larger circle with a radius of 3.

By examining these equations within the context of the given inequalities, you can visualize and sketch the region — a common skill in plane geometry, helping solve complex problems more intuitively.