Problem 78 - Algebra and Trigonometry

Question

77–80 ? Sketch the region given by the set. $$ \left\\{(x, y) | x^{2}+y^{2} \leq 1\right\\} $$

Step-by-Step Solution

Answer

Sketch a filled circle centered at the origin with radius 1.

Step 1: Understand the Inequality

The given inequality describes the set of points \((x, y)\) such that the sum of their squares is less than or equal to 1. Mathematically, \(x^2 + y^2 \leq 1\). This is the equation of a circle centered at the origin \((0, 0)\) with a radius of 1.

Step 2: Identify the Region

Since the inequality is \(\leq\), it means we include all the points on or inside the circle of radius 1. So, the region is a filled disk, including the boundary.

Step 3: Sketch the Circle

Draw a coordinate plane. Plot a circle centered at the origin with radius 1. This can be done by marking points at \((1, 0)\), \((0, 1)\), \((-1, 0)\), and \((0, -1)\), and then connecting these points in a smooth curve.

Step 4: Shade the Interior

Shade the entire region inside the circle, including the circle itself (boundary). This represents the set of all points \((x, y)\) where \(x^2 + y^2 \leq 1\).

Key Concepts

Inequality

In mathematics, inequalities are used to compare expressions and determine the range of possible values that satisfy a given condition. When we look at the inequality \(x^2 + y^2 \leq 1\), we're essentially investigating a set of points that lie in or on a particular geometric shape in the plane.

The "\(\leq\)" sign in the inequality indicates that the solutions include the points exactly on the boundary, as well as those within the boundary. This particular inequality forms a circular region because it describes all points \((x, y)\) where the sum of the squares of \(x\) and \(y\) is less than or equal to 1. Thus, it's not only creating a boundary but also describing everything enclosed within it.

• "\(<\)" would create an open circle, excluding the boundary.
• "\(=\)" would define just the edge of the circle.

In this inequality, the shaded area would prompt us to paint every point inside the circle including its boundary.

Circle

A circle in coordinate geometry represents a simple geometric shape defined by all points equidistant from a fixed center point. In this context, the equation of the circle is \(x^2 + y^2 = 1\). This tells us that the center is at the origin \((0,0)\) and the radius is 1.

The equation \(x^2 + y^2 = r^2\) is a standard equation of a circle, where \(r\) represents the radius. In more generalized terms, if a circle's center is given by \((h, k)\), the equation modifies to \((x-h)^2 + (y-k)^2 = r^2\).

• The boundary of the circle includes all points for which \(x^2 + y^2 = 1\).
• Variations in \(r\) alter the circle's size proportionately.
• Changing the center adjusts the circle's position in the plane.

A circle is always symmetric about its center and thus serves as an essential tool in coordinate geometry for defining regions.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is fundamental in geometry and graphing. It consists of two axes: the horizontal axis, known as the x-axis, and the vertical axis, known as the y-axis. These two intersect at the origin, \((0, 0)\), forming a grid to plot points.

Understanding the coordinate plane is crucial for locating points, drawing shapes, and analyzing their positions. In the circle inequality \(x^2 + y^2 \leq 1\), the coordinate plane helps us sketch and visualize the region we're examining. By marking specific points, such as \((1, 0)\), \((0, 1)\), \((-1, 0)\), and \((0, -1)\), it becomes easier to draw the boundary of a circle.

• The x-axis and y-axis divide the plane into four quadrants.
• Quadrant I: both x and y are positive.
• Quadrant II: x is negative, y is positive.
• Quadrant III: both x and y are negative.
• Quadrant IV: x is positive, y is negative.

Once the circle boundary is drawn, shading inside helps represent the solution to the inequality, showing all included points within the defined region.