Problem 33 - College Algebra

Question

Write the standard form of the equation of the circle with the given center and radius. $$ \text { Center }(3,2), r=5 $$

Step-by-Step Solution

Answer

The standard form of the equation for the circle with center \((3,2)\) and radius 5 is \((x - 3)^2 + (y - 2)^2 = 25\).

Step 1: Identify the given center and radius

From the problem, the center of the circle is given as \((3,2)\) and the radius as 5. This implies that \(h = 3\), \(k = 2\), and \(r = 5\).

Step 2: Substitute the values into the formula

Substitute these values into the standard equation of the circle, \((x - h)^2 + (y - k)^2 = r^2\). \((x - 3)^2 + (y - 2)^2 = 5^2\).

Step 3: Simplify the equation

\((x - 3)^2 + (y - 2)^2 = 25\). This is the equation of the circle in standard form.

Key Concepts

Standard Form

When dealing with the equation of a circle, the standard form plays a crucial role. The basic equation in standard form is expressed as \((x - h)^2 + (y - k)^2 = r^2\). This formula represents all the points

• \(x\) and \(y\) correspond to the coordinates of any point on the circle.
• \((h, k)\) is the center of the circle.
• \(r\) is the radius of the circle.

The term \((x - h)^2 + (y - k)^2\) tells us how far a point \((x, y)\) is from the center \((h, k)\). We square this distance to ensure it is positive.
For any circle, the perfect match of these elements creates an equation representing every point on its boundary. When substituting specific values for \(h\), \(k\), and \(r\) into the equation, you will end up with an equation that characterizes the circle uniquely.

Center of a Circle

The center of a circle is a fundamental concept as it denotes the midpoint of the circle. In the standard form equation \((x - h)^2 + (y - k)^2 = r^2\), \((h, k)\) represents the center of the circle.
Here are some key points to remember about the center:

• The center \((h, k)\) serves as a reference point. Every point on the circle is equally distant from this center.
• The coordinates \(h\) and \(k\) also tell us the horizontal and vertical shifts from the origin—often considered as a starting reference.

In our original problem, the center was given as \((3, 2)\). This means the circle shifts from the origin three units to the right and two units up on a Cartesian plane.Understanding the center's exact position is vital for sketching the circle correctly and for performing geometric transformations.

Radius

The radius is another core aspect of a circle's equation, and it measures the distance from the center to any point on the circle's outline. In the standard form of a circle's equation \((x - h)^2 + (y - k)^2 = r^2\), \(r\) stands for the radius.
Let's go over some important characteristics:

• The radius is always a positive value. Even when squared, it remains positive.
• In graphing terms, the radius determines the size of the circle. A larger radius results in a bigger circle.

In the given exercise, the radius was specified as 5. This radius ensures that the circle's edge is precisely 5 units away from its center at any direction on the plane. By squaring the radius \(5^2 = 25\), we are able to complete the equation in its standard form, providing a correct description of the circle. Grasping the concept of the radius supports a better overall understanding of a circle's geometry and spatial relationships.