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Chapter 2: Problem 10

(a) find the center-radius form of the equation of each circle, and (b) graph it. center \((-3,-2),\) radius 6

Short Answer

Expert verified
(x + 3)^2 + (y + 2)^2 = 36

Step by step solution

01

Understand the Center-Radius Form

The center-radius form of a circle's equation is given by \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center of the circle and \( r \) is the radius.
02

Identify the Given Information

Here, the center of the circle is \( (-3, -2) \) and the radius is \( 6 \). So \( h = -3 \), \( k = -2 \), and \( r = 6 \).
03

Substitute into the Formula

Substitute the given center and radius into the center-radius form equation: \((x - (-3))^2 + (y - (-2))^2 = 6^2\).
04

Simplify the Equation

Simplify the expression to get the center-radius form:\( (x + 3)^2 + (y + 2)^2 = 36 \).
05

Graph the Circle

To graph the circle, plot the center at \(-3, -2\) on the coordinate plane. Mark points that are 6 units away from the center in all directions (up, down, left, right). Draw a smooth curve through these points to complete the circle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Center-Radius Form
The center-radius form is an easy way to describe a circle on a coordinate plane. This form is written as \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) represents the center of the circle, and \( r \) stands for the radius. Knowing this form is essential because it breaks down the relationship between the circle's center and radius using simple algebra. For example, given a circle with center at (-3, -2) and radius 6, we substitute into the formula to get \((x + 3)^2 + (y + 2)^2 = 36 \). This equation tells us everything we need to know about the circle's size and location.
Graphing Circles
Graphing a circle is straightforward once you know its center and radius.
Here are the steps to follow:
  • Plot the center of the circle on the coordinate plane. In our example, the center is at (-3, -2).
  • From the center, measure the radius (6 units in this case) in all four cardinal directions: up, down, left, and right.
  • Plot these four points. These points are located at (-3, 4), (-3, -8), (3, -2), and (-9, -2).
  • Draw a smooth curve through these points to form a circle.
Coordinate Plane
The coordinate plane is crucial for accurately plotting geometric shapes like circles. It consists of two perpendicular lines called the x-axis (horizontal) and y-axis (vertical), which intersect at the origin (0,0).
Each point on the plane is identified by a pair of coordinates (x, y) showing its position relative to the origin. For circles, the coordinate plane helps us:
  • Locate the center of the circle accurately.
  • Measure distances like the radius from the center to any point on the circumference.
  • Visualize the symmetry and roundness of the shape.

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Most popular questions from this chapter

Use a graphing calculator to solve each linear equation. $$3(2 x+1)-2(x-2)=5$$

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