Problem 40 - Algebra A Combined Function

Question

Write an equation of the circle with the given center and radius. See Example 8. $$ (-7,6) ; 2 $$

Step-by-Step Solution

Answer

The equation is \\ (x + 7)^2 + (y - 6)^2 = 4 \\.

Step 1: Understand the Circle Equation

The standard form of the equation of a circle with center \(h, k\) and radius \ r \ is given by the expression: \[ (x - h)^2 + (y - k)^2 = r^2 \]Here, \(h, k\) represents the coordinates of the center and \ r \ is the radius of the circle.

Step 2: Identify Given Values

From the exercise, the center of the circle is \ (-7, 6) \ and the radius is \ 2 \. This means \(h = -7\), \(k = 6\), and \(r = 2\).

Step 3: Substitute Values into the Circle Equation

Substitute \(h = -7\), \(k = 6\), and \(r = 2\) into the standard equation of a circle:\[ (x + 7)^2 + (y - 6)^2 = 2^2 \]This gives:\[ (x + 7)^2 + (y - 6)^2 = 4 \]

Step 4: Write the Final Equation

The equation of the circle with center \ (-7, 6) \ and radius \ 2 \ is:\[ (x + 7)^2 + (y - 6)^2 = 4 \]

Key Concepts

Circle Standard Form

The standard form of a circle's equation is a neat way to express all of its essential properties. This form is written as: \[ (x - h)^2 + (y - k)^2 = r^2 \] In this equation:

• \(x\) and \(y\) are variables representing coordinate points on the circle.
• \(h\) and \(k\) denote the fixed coordinates of the circle's center.
• \(r\) is the radius, the distance from the center to any point on the circle.

This equation allows you to quickly see where the center of the circle is and how large it is. By simply rearranging or filling in these variables, you can describe any circle mathematically. Remember, both \(x\) and \(y\) terms are squared to incorporate all points along the circumference.

Coordinates of the Center

The coordinates of a circle's center are crucial as they pinpoint the exact position of the circle in the coordinate plane. In our standard equation \((x - h)^2 + (y - k)^2 = r^2\), the center coordinates are represented by \(h\) and \(k\). For example, if you are given the center \((-7, 6)\), this tells you:

• The \(x\)-coordinate of the center is \(-7\).
• The \(y\)-coordinate of the center is \(6\).

These values get plugged directly into the standard form equation, adjusting how \(x\) and \(y\) are transformed and offset from the origin. Knowing these coordinates ensures that you can correctly place the circle on the graph, aligning it perfectly with your mathematical model.

Radius of the Circle

The radius of a circle is the distance from the center to any point on the circle's perimeter. It is symbolized by \(r\) in the standard equation. The radius is significant as it determines the size of the circle. In our given example, the radius is \(2\). This means:

• From the center point, you can measure 2 units outward to reach the circle edge.
• It is squared in the equation to \(r^2 = 2^2\), which becomes \(4\).

By inputting this value into the equation, \[(x + 7)^2 + (y - 6)^2 = 4\], it expresses how far the circle extends in all directions from the center. Remember, the radius defines the scale of the circle's spread, rendering it fundamental to size and shape discussions.