Problem 74 - College Algebra with Modeling and Visualization

Question

Find the standard equation of a circle that satisfies the conditions. Radius \(1,\) center \((0,0)\)

Step-by-Step Solution

Answer

The equation of the circle is \\(x^2 + y^2 = 1\\).

Step 1: Identify the Standard Form of a Circle Equation

The standard equation of a circle with its center at \(h, k\) and radius \ r \ is given by \((x - h)^2 + (y - k)^2 = r^2\).

Step 2: Substitute the Known Values

In this case, the center of the circle is \(0, 0\) and the radius \(r\) is \ 1\. Substitute these values into the standard equation. This gives \((x - 0)^2 + (y - 0)^2 = 1^2\).

Step 3: Simplify the Equation

Simplify the equation by eliminating the zeros, resulting in \(x^2 + y^2 = 1\).

Key Concepts

Radius

The radius of a circle is a measurement of the distance from the very center of the circle to any point lying on the circle itself. It’s like drawing a line from the midpoint straight to the outer edge. This distance being constant is what forms the perfect round shape that is a circle.

• The radius is half the diameter. If you know the diameter, just divide it by 2 to find the radius.
• The symbol for the radius is often represented by the letter r.

In the context of our original problem, the radius is given as 1 unit. The formula shows that wherever you travel from the center, you will reach the edge in exactly one unit.

Center of a Circle

The center of a circle is a fixed point within the circle from which every point on the perimeter is equidistant, exactly the length of the radius away.

• The center is denoted by coordinates \(h, k\), which tell you where the center point is located on the Cartesian plane.
• In a circle's equation, these coordinates help define the precise spot in the grid where the center lies.

For our particular problem, the center of the circle is located at the origin, \(0, 0\). This is a special circumstance where the center sits at the very heart of the graph grid, with equal radius extending in all directions.

Standard Form of a Circle

The standard form of a circle's equation is an algebraic expression that neatly describes a circle in terms of its center and radius. This form is very helpful as it allows any circle to be interpreted and easily manipulated: \[(x - h)^2 + (y - k)^2 = r^2\]

• Here, \(h, k\) represents the center of the circle.
• The \(r\) stands for the radius.

By substituting the center and radius into the standard formula, it defines the circle precisely.
In this exercise, after placing our center at \(0, 0\) and plugging the radius, we simplify \[(x - 0)^2 + (y - 0)^2 = 1^2\] to get \(x^2 + y^2 = 1\), explaining a circle centered at the origin with a radius of 1.