Problem 37 - Algebra A Combined Function

Question

Without graphing, how can you tell that the graphs 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 but have different radii, so they do not intersect.

Step 1: Identify the Equations

The given equations are \(x^{2} + y^{2} = 1\) and \(x^{2} + y^{2} = 4\). Both of these equations represent circles.

Step 2: Understand Each Circle

The equation \(x^{2} + y^{2} = 1\) represents a circle centered at the origin \((0, 0)\) with a radius of 1. The equation \(x^{2} + y^{2} = 4\) represents another circle centered at the origin \((0, 0)\) but with a larger radius of 2.

Step 3: Recognize the Relationship

Since both circles are centered at the origin, their centers are the same. However, because the radii of the circles are different, these circles do not overlap.

Step 4: Check the Distance Criteria

If the distance between the centers of two circles is zero and the sum of their radii is greater than the distance, then the circles intersect. In this case, the sum of the radii is \(1 + 2 = 3\), and since they have the same center, the distance between the centers is 0, which means the smaller circle is completely inside the larger circle without touching it.

Step 5: Conclusion

Because one circle lies entirely within the other without touching the larger circle's boundary, there are no points of intersection between the two circles.

Key Concepts

Intersection of Circles

The intersection of circles is an interesting concept in geometry, as it helps us visually and mathematically determine the points where two circles meet. To check if two circles intersect, we can use the equation of each circle to understand their positions relative to one another.
For the circles given by the equations \(x^2 + y^2 = 1\) and \(x^2 + y^2 = 4\), both have the same center at \((0,0)\). This hints that the possible interaction is purely based on their sizes or radii.

• If two circles intersect, the distance between their centers should be less than the sum and more than the difference of their radii.
• If the distance equals the sum of the radii, the circles are tangent to each other, meaning they touch at just one point.
• Zero distance between centers implies one circle is inside the other without any intersection along their circumferences.

Understanding these relationships and attributes helps in quickly identifying possible intersections without drawing the circles.

Circle Radius

The radius of a circle is one of its defining features and plays a crucial role in determining the size and position of the circle. When you have an equation of a circle, like \(x^2 + y^2 = r^2\), the value of \(r\) is the radius. For both given equations, the values of \(r\) help us determine how these circles might interact.

• In the first circle's equation, \(x^2 + y^2 = 1\), the radius is \(r = 1\).
• For the second circle, \(x^2 + y^2 = 4\), the radius is \(r = 2\).

The importance of the radius becomes evident when we try to analyze relationships like intersection or containment between the circles. A larger radius means a comparatively larger circle, affecting how these circles might meet or interact.

Distance Between Centers

The distance between centers of two circles, especially when their centers are the same, simplifies our calculations drastically. For our circles, their centers are both located at \((0, 0)\), making the distance between them zero.

• If the distance was greater than zero, we would use the distance formula: \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \) to determine how far apart the centers are.
• A zero distance means any interaction purely depends on the radii.

Understanding the distance between circles and how it relates to their radii is crucial to predicting circle behavior – such as if they'll intersect, overlap, or remain separate. In our case, the zero distance implies that knowing just the radius tells us one circle is wholly within the other.

Non-Intersecting Circles

Non-intersecting circles often mean that one circle lies fully inside the other without actually meeting its boundary – typically, this adequates to having the smaller circle's entirety positioning inside the larger one's circumference. This occurs when our conditions regarding distances and radius do not allow any touching or overlapping of boundaries.
In our situation, where one circle has a radius of \(1\) and the other has a radius of \(2\), and both share the center, there are some clear pointers:

• The sum of the radii \((1 + 2 = 3)\) completely envelops the smaller circle inside the larger, but no common points touch the boundary.
• Since no part of the smaller circle's boundary reaches the larger one's, they aren't intersecting.

Recognizing non-intersecting circles effectively, through the use of the radii and understanding of distances, guides predictions on their interaction outcome – clear from these kinds of exercises and contributes to a broader recognition of circle behaviors.