Problem 31 - Algebra 2 and Trigonometry

Question

Airplane passengers have been surprised to look down over farmland and see designs, called crop circles, cut into cornfields. Suppose a farmer wishes to make a design consisting of three concentric circles, that is, circles with the same center but different radiu. Write the equations of three concentric circles centered at a point in a cornfield with coordinates \((2,2) .\)

Step-by-Step Solution

Answer

The equations are \((x - 2)^2 + (y - 2)^2 = 9\), \((x - 2)^2 + (y - 2)^2 = 25\), and \((x - 2)^2 + (y - 2)^2 = 49\).

Step 1: Understand the Problem

We need to write equations for three concentric circles centered at the point \((2,2)\). A concentric circle means that all circles share the same center but have different radii.

Step 2: General Equation of a Circle

The standard equation for a circle with center \((h,k)\) and radius \(r\) is given by \[(x - h)^2 + (y - k)^2 = r^2\].For circles centered at \((2,2)\), this becomes \[(x - 2)^2 + (y - 2)^2 = r^2\].To find the equations, we need to choose different values for \(r\).

Step 3: Choose Radii for the Circles

Select three different radii for the concentric circles. For instance, let's choose radii of 3, 5, and 7. These values are arbitrary but distinct, ensuring that the circles do not overlap entirely.

Step 4: Write the Equation for the First Circle

For a radius of 3, the circle's equation is:\[(x - 2)^2 + (y - 2)^2 = 3^2\].Simplifying gives:\[(x - 2)^2 + (y - 2)^2 = 9\].

Step 5: Write the Equation for the Second Circle

For a radius of 5, the circle's equation is:\[(x - 2)^2 + (y - 2)^2 = 5^2\].Simplifying gives:\[(x - 2)^2 + (y - 2)^2 = 25\].

Step 6: Write the Equation for the Third Circle

For a radius of 7, the circle's equation is:\[(x - 2)^2 + (y - 2)^2 = 7^2\].Simplifying gives:\[(x - 2)^2 + (y - 2)^2 = 49\].

Key Concepts

Equation of a Circle

The equation of a circle is a fundamental concept in geometry, describing the set of all points that maintain a constant distance from a fixed central point. This distance is known as the circle's radius. The standard form of a circle's equation is

• The center is located at the point \((h, k)\).
• The constant distance from this center is the radius \(r\).

Substituting this information into the circle equation leads us to:\((x - h)^2 + (y - k)^2 = r^2\).
This formula helps in expressing and determining the location and size of a circle in a coordinate space. By plugging in values for the circle's center and corresponding radius, you can easily construct a specific circle's equation. For example, if the center is at \((2, 2)\) and you have varying radii, you can derive multiple circle equations like \((x - 2)^2 + (y - 2)^2 = r^2\), where \(r\) could be any chosen radius, such as 3, 5, or 7.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses a coordinate system to explore and describe the properties of geometric shapes. It's a bridge between algebra and geometry through graphs and equations. This approach helps understand geometric figures in terms of coordinates. For circles, this involves working with the circle equations and plotting them in a two-dimensional plane.
In our case of concentric circles:

• All circles share the same center, which is represented by a point on the coordinate plane, specifically \((2, 2)\).
• The varying radii lead to circles of different sizes, but they all center around the same point.

Utilizing coordinate geometry allows for precise plotting and visualization of these relationships, making it possible to see how different radii produce separate, yet related, circles.

Circle Radii

Radii are a crucial aspect of circle geometry. They determine the size of circles and contribute to understanding their properties. The radius is the line segment from the center to any point on the circle's circumference. In the context of concentric circles:

• Multiple circles share a common center but have different radii.
• These radii help distinguish each circle in terms of size.

Choosing distinct radii like 3, 5, and 7 ensures each circle remains unique and identifiable, while still part of the concentric arrangement. Calculating radii helps to specify the circle equations as well, as seen in expressions such as: \((x - 2)^2 + (y - 2)^2 = 9\), \((x - 2)^2 + (y - 2)^2 = 25\), and \((x - 2)^2 + (y - 2)^2 = 49\).
These showcase how the circles expand outward symmetrically with their singular but shared center.