Problem 28 - A Graphical Approach to College Algebra

Question

Write an inequality that satisfies the description. Outside the circle with radius 3 and center ( \(0,0\) )

Step-by-Step Solution

Answer

The inequality is \(x^2 + y^2 > 9\).

Step 1: Understand the Description

We need to find an inequality that describes all the points located outside a circle centered at the origin (0,0) with a radius of 3.

Step 2: Recall the Equation of a Circle

The equation for a circle with center \(h,k\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). For a circle centered at (0,0) with radius 3, the equation is \((x - 0)^2 + (y - 0)^2 = 3^2\), simplifying to \(x^2 + y^2 = 9\).

Step 3: Convert the Equation to an Inequality

To describe points outside the circle, we use an inequality. The inequality that represents points outside the circle with equation \(x^2 + y^2 = 9\) is \(x^2 + y^2 > 9\). This inequality means points have a greater distance from the origin than 3 (the radius).

Key Concepts

Circles

A circle is a set of points in a plane that are equidistant from a given point called the center. The distance from the center to any point on the circle is defined as the radius. In our example, the circle is centered at the origin (0,0) and has a radius of 3.

To find whether a point is inside, on, or outside a circle, we use its equation. The equation of a circle is given by \( (x - h)^2 + (y - k)^2 = r^2 \), where (h,k) represents the center, and r is the radius.

In our case, since the center is at (0,0), the equation simplifies to \( x^2 + y^2 = 9 \). All points (x, y) that satisfy this equation lie on the circle. For points outside, you adjust the equation into an inequality such as \( x^2 + y^2 > 9 \), which signifies that the points are farther from the center than the radius.

Coordinate System

The coordinate system is a grid that helps us locate points in a plane through an ordered pair (x,y). The coordinates refer to positions along the x and y axes, which intersect at the origin (0,0). This system makes it easy to denote locations of points, lines, and shapes like circles.

Understanding coordinates is essential for defining geometric figures and their properties. In our context, knowing that the circle is centered at (0,0) helps in writing the circle's equation directly. The origin becomes a pivotal point, allowing us to focus on distances relative to it when solving problems involving circles and distances.

It's important to note that distances in a coordinate system traditionally utilize the Euclidean method, which forms the basis for our third key concept.

Distance Formula

The distance formula is key in geometry, particularly when working with circles in a coordinate system. It calculates the distance between two points, say (x_1, y_1) and (x_2, y_2), using this formula:

\[ \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

For a circle, this formula determines the distance from the center to any point on or outside the circle. If the calculated distance is equal to the radius, the point is on the circle. If greater, then the point lies outside.

In the original exercise, the task was to recognize that for points outside the circle of radius 3 centered at (0,0), the inequality \( x^2 + y^2 > 9 \) represents all such points. Here, the distance from each point to the origin must be greater than 3, aligning with the distance formula where the radius acts as a benchmark.