Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Find the center-radius form of the equation of a circle with the given center and radius. Graph the circle. Center \((0,0),\) radius 2

Step-by-Step Solution

Answer

The circle's equation is \(x^2 + y^2 = 4\).

Step 1: Understanding the General Equation of a Circle

The general equation of a circle in center-radius form is \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is the radius.

Step 2: Substitute the Center into the Equation

The center of the circle is given as \((0, 0)\), so substitute \(h = 0\) and \(k = 0\) into the equation to get \((x-0)^2 + (y-0)^2 = r^2\).

Step 3: Substitute the Radius into the Equation

The radius is given as \(2\). Substitute \(r = 2\) into the equation, resulting in \((x-0)^2 + (y-0)^2 = 2^2\).

Step 4: Simplify the Equation

Simplify the equation by eliminating zero terms: \(x^2 + y^2 = 4\).

Step 5: Graph the Equation

To graph the equation \(x^2 + y^2 = 4\), plot a circle on the coordinate plane with center at \((0, 0)\) and a radius of 2.

Key Concepts

Center-Radius Form

The center-radius form of a circle's equation is a powerful and common method used in geometry to express circles on a coordinate plane. This form allows us to define a circle by knowing just two elements: its center and its radius.

In the center-radius form, the equation of a circle is written as:

• \[(x-h)^2 + (y-k)^2 = r^2\]

- Here, \(h\) and \(k\) represent the x and y coordinates of the circle’s center, respectively.- \(r\) is the radius of the circle.
When a circle’s center is located at the origin, like in our example (\((0,0)\)), the equation becomes simpler:

• \[x^2 + y^2 = r^2\]

This form is convenient as it directly illustrates the relationship between the center, radius, and any point \(x,y\) on the circle. Understanding this relationship allows one to not only write the equation of a circle but also to analyze its properties effectively.

Graphing Circles

Graphing circles using the center-radius form is a straightforward process once you understand the layout of the coordinate plane and the properties of the circle. Given an equation like \[x^2 + y^2 = 4\], here's how you can graph it:

• Start by plotting the center of the circle. In our example, the center is at \((0,0)\), which is the origin.
• The next step is to use the radius. From the equation \[x^2 + y^2 = r^2\], we know \(r^2\) is 4, which means the radius is 2.
• Measure 2 units from the center in all directions. This gives us key points that the circle will pass through, such as \((2,0)\), \((-2,0)\), \((0,2)\), and \((0,-2)\).
• Draw a smooth curve connecting these points to complete the circle.

Circles are symmetric along both axes, which makes graphing them visually intuitive once you place these directional guide points.

Coordinate Plane

The coordinate plane is an essential tool in geometry and algebra that provides a visual framework for graphing functions and shapes like circles. It consists of two perpendicular axes:

• The horizontal axis, known as the x-axis, and
• The vertical axis, known as the y-axis.

Where these axes intersect is called the origin, denoted as \((0,0)\).

On this plane, each point can be described using a pair of numerical values, referred to as coordinates. The first value corresponds to the x-axis, and the second to the y-axis, like \((x, y)\).

Using the coordinate plane to graph a circle means ensuring that all points \((x,y)\) that satisfy the circle's equation \[(x-h)^2 + (y-k)^2 = r^2\] are accounted for.

This graphical method facilitates a clear visual understanding of geometric relationships, making the concepts more tangible.