Problem 26 - Beginning and Intermediate Algebra

Question

Find an equation of the circle with the given center and radius. Center \((0,-3) ;\) radius \(=5\)

Step-by-Step Solution

Answer

The equation of the circle with center (0, -3) and radius 5 is: \[(x^2) + (y+3)^2 = 25\]

Step 1: Identify center and radius

The center of the circle is at point (0, -3) and the radius is 5 units.

Step 2: Apply the standard equation for a circle

We will use the equation \[(x-a)^2 + (y-b)^2 = r^2\], where (a, b) is the center of the circle and r is the radius. In this case, a = 0, b = -3, and r = 5.

Step 3: Plug in the known values into the equation

Now, substitute the values a, b, and r into the equation: \[(x-0)^2 + (y-(-3))^2 = 5^2\] Simplify to get the equation of the circle: \[(x^2) + (y+3)^2 = 25\]

Step 4: Final equation of the circle

The equation of the circle with center (0, -3) and radius 5 is: \[(x^2) + (y+3)^2 = 25\]

Key Concepts

Center of a Circle

The center of a circle is a crucial point that serves as the middle of the circle. In coordinate geometry, it is represented by a pair of coordinates \(a, b\). Each point on the circle is equidistant from this central point.
This unique property defines all points belonging to the circle. For example, if the center is (0,-3), this means that every point on the circle is exactly the same distance from the position at x=0 and y=-3.
The center provides a point of reference for equations and helps in graphing circles easily on a coordinate plane. Knowing the center allows you to precisely locate the circle in a given coordinate system.

Radius of a Circle

The radius of a circle is the distance from the center to any point on the circle. This distance remains constant no matter where you measure on the circle's perimeter.
Think of it as the invisible line that connects the center to the edge of the circle. In mathematical terms, the radius is a key variable that helps define the circle's size and area.
When given a radius like 5 units, it's saying the stretch from center to edge is exactly 5 units.

• The radius is often denoted by the variable \(r\).
• It is half of the diameter, which is the total distance across the circle passing through the center.
• In circle equations, the radius appears squared, as it relates to the radius squared (\(r^2\)).

By knowing the radius, you can calculate other properties such as the area and circumference, offering a comprehensive understanding of the circle's dimension.

Standard Form of a Circle

The standard form of a circle's equation is fundamental in mathematics for representing circles on a graph. The general expression is \( (x-a)^2 + (y-b)^2 = r^2 \).
This equation effectively ties together the concepts of center (\(a,b\)) and radius (\(r\)).

• \(a\) and \(b\) are the x and y coordinates of the center.
• \((x-a)^2\) and \( (y-b)^2\) denote the squared distances from a particular point \( (x,y) \) on the circle back to the center.
• The \(r^2\) represents the square of the radius, ensuring the equation evaluates to the radius's length squared.

If you substitute the values of the center and radius into this formula, it precisely locates the entire circle. For example, substituting \(a=0\), \(b=-3\), and \(r=5\) into the formula gives \(x^2 + (y+3)^2 = 25\). The standard form is essential as it provides a straightforward way to graph and analyze circles algebraically.