Problem 18 - Algebra A Combined Function

Question

The graph of each equation is a circle. Find the center and the radius, and then graph the circle. See Examples 5 through 7. $$ x^{2}+y^{2}=25 $$

Step-by-Step Solution

Answer

Center: (0, 0); Radius: 5

Step 1: Identify the Standard Equation of a Circle

The standard equation of a circle with center \((h, k)\) and radius \(r\) is given by \((x-h)^{2}+(y-k)^{2}=r^{2}\). In the given equation, \(x^{2}+y^{2}=25\), compare it with the standard form.

Step 2: Determine the Center of the Circle

The given equation is \(x^{2}+y^{2}=25\). Since there are no \(h\) and \(k\) terms, it implies \((h, k) = (0, 0)\). Thus, the center of the circle is at the origin.

Step 3: Calculate the Radius of the Circle

Compare the given equation \(x^{2}+y^{2}=25\) with the standard form \((x-h)^{2}+(y-k)^{2}=r^{2}\). Here, \(r^{2}=25\). Taking the square root of both sides, \(r = \sqrt{25} = 5\). Hence, the radius of the circle is 5.

Step 4: Graph the Circle

To graph the circle, plot the center at (0, 0). From this point, mark the radius distance (5 units) in all directions: left, right, up, and down. Sketch the circle by connecting these points with a smooth curve.

Key Concepts

Center of a Circle

When discussing circles in the coordinate plane, the word "center" refers to a specific point from which all points on the circle are equidistant. In mathematical terms, if the center is at the point \(h, k\), then every point on the circle is equally distant from this central point.
In the standard equation \( (x-h)^2 + (y-k)^2 = r^2 \), the coordinates \(h\) and \(k\) represent the coordinates of the center. \(h\) is the x-coordinate, while \(k\) is the y-coordinate.
For the equation \(x^2 + y^2 = 25\), we observe there is no \(h\) or \(k\) term present. This absence signifies that both \(h\) and \(k\) are zero. Therefore, the center of this circle is at the origin:

• Center = \( (0, 0) \)

Understanding how to locate the center helps in constructing and visualizing the circle on a graph with accuracy.

Radius of a Circle

The radius of a circle is the straight-line distance from the center of the circle to any point on its circumference. This distance is constant for all circles.
The radius is represented by the variable \(r\) in the equation \( (x-h)^2 + (y-k)^2 = r^2 \). In the circle equation \(x^2 + y^2 = 25\), we can compare it with the standard form to identify \(r^2 = 25\).
By solving for \(r\), we take the square root of 25, giving us \(r = \sqrt{25} = 5\). Hence, the radius of the circle is:

• Radius = 5 units

This constant value ensures that each point on the circle is 5 units away from the center, maintaining the circle's perfect shape.

Standard Form of Circle Equation

The standard form of a circle equation is a concise way to express the relationship between the coordinates of points on a circle, its center, and its radius. The format of this equation is:
\((x-h)^2 + (y-k)^2 = r^2 \)
Here:

• \(h\) and \(k\) pinpoint the center of the circle
• \(r\) is the radius

The standard equation clearly shows how a circle shifts across a coordinate plane as its center changes from the origin.
In our example, \(x^2 + y^2 = 25\), since it matches the form where both \(h\) and \(k\) are zero, the discovery is:

• Center: Origin \( (0, 0) \)
• Radius: Calculated as \( r = \sqrt{25} = 5 \)

Using this form makes it easier to graph and understand the geometry of the circle based on its equation.