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Chapter 10: Problem 48

Find the center and the radius of each circle. Then graph the circle. $$(x-2)^{2}+(y+3)^{2}=100$$

Short Answer

Expert verified
Center: (2, -3), Radius: 10

Step by step solution

01

Identify the Standard Form of the Circle

The equation \((x-2)^2 + (y+3)^2 = 100\) is in the standard form of a circle, which is \((x-h)^2 + (y-k)^2 = r^2\).
02

Determine the Center

In the standard form \((x-h)^2 + (y-k)^2 = r^2\), \(h\) and \(k\) represent the coordinates of the center of the circle. Here, \h = 2\ and \k = -3\. So, the center of the circle is \((2, -3)\).
03

Find the Radius

The constant on the right side of the standard form equation represents \r^2\. In this equation, \(r^2 = 100\). To find the radius, take the square root of 100: \r = \sqrt{100} = 10\. Thus, the radius of the circle is 10 units.
04

Graph the Circle

To graph the circle, plot the center at \(2, -3)\). From the center, measure and mark points 10 units away in all directions (up, down, left, and right). These marks will help to sketch the circle accurately.

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Key Concepts

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

Standard Form of a Circle
Understanding the standard form of a circle is crucial. The standard form of a circle equation is expressed as \((x-h)^2 + (y-k)^2 = r^2\). Here:
  • \(h\) and \(k\) are the x and y coordinates of the circle's center.
  • \(r\) is the radius of the circle.
This standard format makes it easier to identify the key components of the circle 鈥� its center and its radius 鈥� directly from the equation. If you see any equation in this form, you can immediately spot the center \(h, k\) and the radius \(r\) by extracting these values.
Let鈥檚 parse an example: For the equation \((x-2)^2 + (y+3)^2 = 100\), it aligns perfectly with the standard form.
Center of a Circle
The center of a circle in the standard form equation \((x-h)^2 + (y-k)^2 = r^2\) can be easily identified. The terms \(h\) and \(k\) represent the x and y coordinates of the center. Therefore, if our equation is \((x-2)^2 + (y+3)^2 = 100\),
We observe:
  • \(h = 2\)
  • \(k = -3\)
So, the center of this circle is at the coordinates \((2, -3)\).
It helps to remember this simple rule:
  • Subtracting a number inside the parenthesis means the center moves to the positive direction of that axis.
  • Adding a number means it moves to the negative direction of that axis.
This is a reliable method to determine the center coordinates in any given circle equation.
Radius of a Circle
In the standard form equation \((x-h)^2 + (y-k)^2 = r^2\), \(r\) stands for the radius of the circle. To determine the radius, take the square root of the constant on the right-hand side.
Let's break it down with our example:
  • Equation: \((x-2)^2 + (y+3)^2 = 100\)
  • Here, \(r^2 = 100\)
  • To find \(r\), we take the square root of 100
  • Thus, \(r = \sqrt{100} = 10\)
Therefore, the radius of the circle is 10 units. Always ensure to square root the number carefully to find the correct radius.
Graphing Circles
To graph a circle given its standard form equation, follow these steps:
  1. Plot the center located at \(h, k\).
  2. From the center, measure the radius distance in all four main directions: up, down, left, and right.
  3. Mark these points.
  4. Connect these points in a smooth, round shape to complete the circle.
For the equation \((x-2)^2 + (y+3)^2 = 100\), we plot:
  • Center at \(2, -3\)
  • Radius of 10 units
Measure 10 units from \(2, -3\) in each direction to mark the circle's boundary points.
Join these points smoothly to visualize the circle. This care in plotting ensures the circle is accurately represented on the graph.

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