Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form of the equation of a circle with the given center and radius. Graph the circle. Center \((-2,0),\) radius 5

Step-by-Step Solution

Answer

The equation is \((x + 2)^2 + y^2 = 25\).

Step 1: Recognize the Center-Radius Form

The center-radius form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle, and \(r\) is the radius.

Step 2: Plug in the Center and Radius

Substitute the given center \((-2, 0)\) and the radius \(5\) into the formula.Substitute \(h = -2\), \(k = 0\), and \(r = 5\) to get \((x + 2)^2 + (y - 0)^2 = 25.\)

Step 3: Simplify the Equation

Simplify the equation by removing unnecessary terms:\((x + 2)^2 + y^2 = 25\).This is the center-radius form of the circle's equation.

Step 4: Describe How to Graph the Circle

To graph the circle, plot the center at \((-2, 0)\) on the coordinate plane. Then, since the radius is 5, mark points 5 units away from the center in all directions (up, down, left, right, and diagonally) and draw a circle through these points.

Key Concepts

Center-Radius Form

The center-radius form of a circle's equation provides an intuitive way to understand the circle's dimensions and position on the coordinate plane. This form is written as:

• \((x - h)^2 + (y - k)^2 = r^2\)

Here, \((h, k)\) represents the center of the circle, and \(r\) is the radius.
This equation is crucial because it clearly shows the location and size of the circle in terms of its center and radius.
By understanding the variables in this equation, you can easily determine the exact point of the circle's center and the distance from the center to any point on the perimeter, which is the radius. Once you have the center and radius, you can construct the equation, as seen in the example noted where the center is \((-2, 0)\) and the radius is 5. Plug these values into the equation:

• \((x + 2)^2 + y^2 = 25\)

This final expression represents the circle in the center-radius form.

Graphing Circles

Graphing circles on a coordinate plane can be both simple and fun when you know the center and radius. Begin by plotting the center point.In our example, the center is \((-2, 0)\). Mark this point on the plane as your starting position.
Once the center is marked, the next step is to use the radius to find points on the circle. Since the radius is 5, measure 5 units outward from the center in all conceivable directions.
This includes straight up, down, left, right, and along diagonal lines.
Finally, connect these points to create a smooth round shape. Make sure your circle goes equally around the center point. This method ensures your circle will look balanced and accurate. With practice, you’ll graph circles quickly and with ease.

Coordinate Geometry

Coordinate geometry, or analytic geometry, combines algebra and geometry using a coordinate system. In this context, the plane is divided into horizontal \(x\)-axis and vertical \(y\)-axis, forming a grid.
Each point in the plane is identified by ordered pairs \((x, y)\), which are essentially coordinates.
For circles, coordinate geometry plays a crucial role in graphically representing equations like \((x + 2)^2 + y^2 = 25\). Here, you use coordination to move precisely through the plane, knowing where to place the center and how far to stretch out the radius.
Using this system, various geometric shapes—including circles—can be graphically represented and manipulated. This integration allows us to solve complex problems by merging algebraic expressions with geometric figures. Essentially, coordinate geometry paves the way for deeper understanding and visual learning in mathematics. It makes abstract mathematical concepts more tangible and simpler to grasp.