Problem 39 - Algebra A Combined Function

Question

Write an equation of the circle with the given center and radius. See Example 8. $$ (2,3) ; 6 $$

Step-by-Step Solution

Answer

The equation of the circle is \((x - 2)^2 + (y - 3)^2 = 36\).

Step 1: Identify the General Equation of a Circle

The general equation of a circle with center \((h, k)\) and radius \(r\) is given by \((x - h)^2 + (y - k)^2 = r^2\). This is the formula we will use to find the equation of the circle.

Step 2: Substitute the Center Values

Substitute the center coordinates \((h, k) = (2, 3)\) into the general equation to get \((x - 2)^2 + (y - 3)^2 = r^2\). This accounts for the circle being centered at \((2, 3)\).

Step 3: Substitute the Radius Value

Substitute the radius \(r = 6\) into the equation to complete the circle's equation. So the equation becomes \((x - 2)^2 + (y - 3)^2 = 6^2\).

Step 4: Simplify the Equation

Simplify the equation by squaring the radius. Since \(6^2 = 36\), the final equation of the circle is \((x - 2)^2 + (y - 3)^2 = 36\).

Key Concepts

Center of a Circle

The center of a circle is an essential point that defines its position in the coordinate plane. In geometry, the center is the fixed point from which every point on the circle is equidistant. This means every point on the circumference of the circle is the same distance from the center.

When working with equations of circles, the center is typically represented by a pair of coordinates, often denoted as \((h, k)\). These coordinates tell us exactly where the center of the circle is located on the grid.

For example, if a problem states the center of a circle is \((2, 3)\), this means that from the origin of the graph, you would move 2 units to the right (if counting along the x-axis), and 3 units up (if counting along the y-axis) to locate the center.

Radius of a Circle

The radius of a circle is a fundamental element. It is the distance from the center to any point on the circle. Essentially, the radius defines the size of the circle - a larger radius means a larger circle, and a smaller radius means a smaller circle.

In an equation of a circle, the radius is denoted by the variable \(r\).

Let's consider a circle with a radius of 6. This means any line segment from the center of the circle, such as \((2,3)\), to any point on the circle will measure 6 units. This distance is always constant for a particular circle.

General Equation of a Circle

The general equation of a circle helps us mathematically express the set of all points equidistant from a center point in the coordinate plane. The equation is usually written as:

• \((x - h)^2 + (y - k)^2 = r^2\)

Here, \((h, k)\) is the center of the circle, and \(r\) is the radius.

This formula allows us to create an algebraic representation of the circle. For example, a center of \((2, 3)\) and radius \(6\) can be inserted into the general equation, resulting in \((x - 2)^2 + (y - 3)^2 = 36\). This tells us the relationship between \(x\) and \(y\), forming a circle in the coordinate plane.

Substitution Method

The substitution method is a straightforward algebraic tool used to replace variables with specific values. This step ensures the general equation accurately reflects the given details of the circle.

To use the substitution method in our case:

• Replace \((h, k)\) with the center coordinates \((2, 3)\).
• Replace \(r\) with the given radius, which is 6.

Substituting these into the general equation \((x - h)^2 + (y - k)^2 = r^2\) turns it into \((x - 2)^2 + (y - 3)^2 = 6^2\).

After this substitution, simplifying further by finding \(6^2\) gives the final equation of the circle as \((x - 2)^2 + (y - 3)^2 = 36\). This substitution and simplification clarify how each element of our circle is accounted for in the equation.