Problem 30 - Intermediate Algebra

Question

Write the equation of a circle in standard form with the following properties. Center at \((0,0) ;\) radius \(\frac{1}{3}\)

Step-by-Step Solution

Answer

The equation is \(x^2 + y^2 = \frac{1}{9}\).

Step 1: Understanding the Problem

We need to write the equation of a circle with a given center and radius. The center of the circle is at \((0,0)\), and the radius is \(\frac{1}{3}\).

Step 2: Circle Equation

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

Step 3: Substitute the Center

Substitute \(h = 0\) and \(k = 0\) since the center of the circle is at the origin. The equation becomes \(x^2 + y^2 = r^2\).

Step 4: Substitute the Radius

Substitute \(r = \frac{1}{3}\) into the equation \(x^2 + y^2 = r^2\). This gives us \(x^2 + y^2 = \left(\frac{1}{3}\right)^2\).

Step 5: Simplify the Radius Squared

Calculate \(\left(\frac{1}{3}\right)^2 = \frac{1}{9}\). The equation then becomes \(x^2 + y^2 = \frac{1}{9}\).

Step 6: Final Equation

The equation of the circle in standard form is \(x^2 + y^2 = \frac{1}{9}\).

Key Concepts

Center of a Circle

The center of a circle is a crucial concept in geometry. It refers to the point around which all points on the circle are equidistant. The center is denoted by the coordinates \(h, k\).
In the given exercise, the center is located at the origin \(0,0\).
This is one of the simplest forms for a circle's center because both the \(h\) and \(k\) values are zero, which simplifies the equation considerably.

• Whenever the center is at the origin, the equation can be simplified to exclude the \(h\) and \(k\) values, leaving a cleaner form.
• This means you just need to focus on the distance from this central point to any point on the circle, which is the radius.

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's denoted by \(r\).
In our case, the radius is given as \(rac{1}{3}\).

• The radius can be any positive number, and it directly affects the size of the circle.
• A smaller radius, like \(rac{1}{3}\), will create a smaller circle.
• Understanding the radius is key to determining how large the circle will be from its center point.

To use the radius in the circle's equation, it is squared. Here, we square \(rac{1}{3}\) to get \(rac{1}{9}\). This final result is plugged into the equation to help define the circle's size and shape properly.

Standard Form of a Circle Equation

The standard form of a circle's equation is a mathematical way to express the circle's dimensions and position. It is represented as \( (x-h)^2 + (y-k)^2 = r^2 \).
This formula highlights several key aspects:

• The \(h\) and \(k\) are the x and y coordinates of the center.
• The \(r\) represents the radius.

By inserting the values for the center and radius, you can specifically describe any circle. In the original problem:

• \(h = 0\), \(k = 0\): due to the center being at the coordinate \(0,0\).
• \(r = rac{1}{3}\): which means the equation becomes \( x^2 + y^2 = rac{1}{9} \).

This form makes it easy to quickly see and understand the circle's features: its center and size. The purpose of this formula is to standardize how circles are represented for easier manipulation and understanding across different mathematical problems.