Problem 41 - Intermediate Algebra

Question

For the exercises below, see the Concept Check in this section. Without graphing, how can you tell that the graph of \(x^{2}+y^{2}=1\) and \(x^{2}+y^{2}=4\) do not have any points of intersection?

Step-by-Step Solution

Answer

The circles are concentric and the smaller is entirely inside the larger, so they don't intersect.

Step 1: Understanding the Equations

The given equations are of the form \(x^2 + y^2 = r^2\), which represent circles centered at the origin (0,0). For \(x^2 + y^2 = 1\), the radius \(r\) is 1. For \(x^2 + y^2 = 4\), \(r\) is 2.

Step 2: Analyzing the Distances

The centers of both circles are at the origin (0,0). Thus, the distance between the centers is 0, which means they are concentric.

Step 3: Comparing Radii

The radius of the smaller circle (\(x^2 + y^2 = 1\)) is 1, and the radius of the larger circle (\(x^2 + y^2 = 4\)) is 2. Since the larger circle's radius is greater than the sum of its distance to the center and the smaller circle's radius, the smaller circle is entirely inside the larger one without touching its boundary.

Step 4: Conclusion on Intersection

Since the smaller circle is completely inside the larger circle without touching, there are no points of intersection between them.

Key Concepts

Intersection of Circles

When we talk about the intersection of circles, we're referring to the points where two circles meet or overlap. To determine if two circles intersect, we need to consider their centers and radii. For instance, if two circles are centered at the same point (concentric circles) and have different radii, like the ones in the given exercise, they may not intersect at all.
If the distance between the centers of the circles is less than the sum of their radii, the circles may intersect at two different points, touch at just one point (if the distance equals the sum of the radii), or not intersect at all if one is completely inside the other. In our scenario, because both circles share the same center and have different radii—one being larger than the other—the smaller circle lies entirely within the larger circle. This spatial arrangement results in no points of intersection.

Distance Between Centers

The distance between the centers of two circles is a crucial factor in determining their position relative to each other. It helps establish whether or not the circles might intersect. For the circles given in the original exercise, both are centered at the origin (0,0), therefore, the distance between their centers is 0.
When circles are concentric, meaning they have the same center, their relative positions only depend on their radii. In such cases, if one circle's radius is greater than the other's, the smaller circle lies entirely within the larger circle without touching its boundary. This scenario confirms there is no point of intersection between the two circles.
In summary, checking the distance between the centers is a straightforward method to see how the circles align and whether or not they intersect.

Radius of a Circle

The radius of a circle is the distance from its center to any point on its boundary. It is an essential component in both identifying a circle's size and determining intersections with other circles. In the problem given, two equations describe circles with radii 1 and 2 units, respectively.
The radius directly impacts how circles overlap or if they even touch. If two circles share the same center, how they compare in size is determined wholly by their radii. The smaller circle, with a radius of 1, falls entirely inside the larger circle with a radius of 2.
Radii can denote full overlap, partial overlap, or no overlap at all. Thus, understanding the radius can guide us to determining the nature of two circles’ intersection—or lack thereof. In this exercise, since the smaller circle does not reach the boundary of the larger one, they conclude without any intersection points.