Problem 24 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

Center: (0, 0); Radius: 5.

Step 1: Recognize the Equation of a Circle

The given equation is \(x^2 + y^2 = 25\). This equation is in the standard form of a circle centered at the origin, \( (0, 0) \), with a radius \( r \). The standard form of a circle is \( (x - h)^2 + (y - k)^2 = r^2 \), where \((h, k)\) is the center and \(r\) is the radius.

Step 2: Identify the Center of the Circle

From the equation \(x^2 + y^2 = 25\), we can see that \(h = 0\) and \(k = 0\) since the equation can be rewritten as \((x - 0)^2 + (y - 0)^2 = 25\). This tells us the center of the circle is at \((0, 0)\).

Step 3: Calculate the Radius of the Circle

The equation \(x^2 + y^2 = 25\) can be compared to \((x - h)^2 + (y - k)^2 = r^2\) to identify that \(r^2 = 25\). To find the radius \(r\), take the square root of 25: \(r = \sqrt{25} = 5\). Thus, the radius of the circle is 5.

Step 4: Graph the Circle

To graph the circle with center \((0, 0)\) and radius 5, plot the center point at the origin on the coordinate plane. Use the radius to measure 5 units in all directions (up, down, left, and right) from the center. Sketch a circle connecting these points, maintaining equal distance from the center all around.

Key Concepts

Graphing Circles

Graphing a circle might sound complicated at first, but it's quite straightforward. Start by understanding the equation of a circle and note whether it’s in the standard form. The standard equation for a circle is \[ \left(x - h\right)^2 + \left(y - k\right)^2 = r^2 \] where \( (h, k) \) denotes the center of the circle and \( r \) is the radius.

To graph a circle effectively:

• Identify the center point \( (h, k) \) on the coordinate plane.
• Determine the radius \( r \), and since you know \( r = 5 \), measure \( 5 \) units from the center in all four cardinal directions (up, down, left, and right).
• Sketch the circle ensuring it's round and symmetric by maintaining the same distance from the center at all points.

A simple way to check your sketch is by ensuring all points on the circle satisfy the original equation. For example, for the circle \( x^2 + y^2 = 25 \), any point \((x, y)\) on the circle should fulfill this equation when substituted back.

Center of a Circle

The center of a circle in the standard circle equation \( \left(x - h\right)^2 + \left(y - k\right)^2 = r^2 \) is denoted by \( (h, k) \). This center point is crucial as it serves as the fixed point from which all points on the circle are equidistant.

For the equation \( x^2 + y^2 = 25 \), rewrite it as \( \left(x - 0\right)^2 + \left(y - 0\right)^2 = 5^2 \). It becomes evident that the values of \( h \) and \( k \) are both zero, placing the center of the circle at the origin, \( (0, 0) \).

Understanding the location of the center helps in accurately placing the circle on a graph. Once the center is pinpointed, you can graph the circle more precisely, using it as your starting point before you sketch the radius outward.

Radius of a Circle

The radius of a circle is a fundamental concept that refers to the distance from the center of the circle to any point on its perimeter. In the standard equation of a circle, \( (x - h)^2 + (y - k)^2 = r^2 \), \( r \) is the radius.

To find \( r \), simply take the square root of the right-hand side of the equation. For the equation \( x^2 + y^2 = 25 \), compare it to the standard form, identifying that \( r^2 = 25 \). Thus, the radius \( r \) is \( \sqrt{25} = 5 \).

This means every point on the circle is exactly \( 5 \) units away from the center \( (0, 0) \). The radius informs the size of the circle: a larger radius results in a bigger circle, while a smaller radius yields a smaller circle.