Problem 42 - College Algebra

Question

Give the center and radius of the circle described by the equation and graph each equation. $$ x^{2}+y^{2}=49 $$

Step-by-Step Solution

Answer

The center of the circle is at the origin (0, 0) and the radius is 7 units.

Step 1: Identify the Center of the Circle

Since the equation is in the standard form \( x^{2}+y^{2}=r^{2} \), the center of the circle is at the origin, (0, 0).

Step 2: Find the radius of the Circle

The constant term on the right hand side of the equation is the square of the radius, i.e., \( r^{2} = 49 \). So, taking the square root of 49 gives us \( r = \sqrt{49} = 7 \). Thus, the radius of the circle is 7 units.

Step 3: Plotting the circle on a graph

To graph the circle, start at the center (0,0). Then plot a point 7 units in all directions - up, down, left, and right. Finally, smoothly connect these points to get a circle.

Key Concepts

Center of a Circle

The center of a circle is the fixed point from which every point on the circumference of the circle is equidistant. In the case of the circle equation \(x^2 + y^2 = 49\), the circle is centered at the origin, which is the point (0, 0). This special positioning occurs when there is no \(x\) or \(y\) term in the equation, indicating that the center is at the coordinate system's starting point.

Understanding the center is crucial because it is the reference point from which you measure all other aspects of the circle, such as the radius, and when graphing the circle.

Radius of a Circle

The radius of a circle is the distance from the center of the circle to any point on its circumference. It is a key dimensional attribute of a circle. From the given equation \(x^2 + y^2 = 49\), the square of the radius, \(r^2\), is equal to 49. Taking the square root gives us the actual radius, \(r = \sqrt{49} = 7\) units.

The radius is fundamental not only to the size of the circle but also to calculations involving area, circumference, and in more advanced contexts, the equation of a circle in Cartesian coordinates.

Graphing Circles

Graphing a circle on the Cartesian plane involves plotting points that are a fixed distance (the radius) from a central point (the center). For the equation \(x^2 + y^2 = 49\), we graph this circle by starting at the center at (0, 0) and moving outwards 7 units in every direction—upwards, downwards, left, and right.

These points represent the north, south, east, and west cardinal points of the circle. Connecting these points with a smooth curve completes the graph of the circle. It is helpful to remember that every point on the circle is exactly 7 units away from the center, creating a perfect symmetrical shape.

Standard Form of a Circle Equation

The standard form of a circle's equation is \(x-h)^2 + (y-k)^2 = r^2\), where \(h, k\) are the coordinates of the center of the circle, and \(r\) is the radius. This form clearly displays the circle's center and radius, which are essential for graphing and understanding the circle's geometry.

For example, in the standard form equation \(x^2 + y^2 = 49\), it can be inferred that \(h=0\) and \(k=0\), as there are no numbers subtracted from \(x\) and \(y\), thus placing the center at the origin. Moreover, \(r^2 = 49\) gives a radius of 7 when solved. Recognizing the standard form aids in quickly identifying a circle's characteristics.