/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 112 Find the center and the radius o... [FREE SOLUTION] | 91影视

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Chapter 1: Problem 112

Find the center and the radius of the circle. Then graph the circle by hand. Check your graph with a graphing calculator: $$(x-7)^{2}+(y+2)^{2}=25$$

Short Answer

Expert verified
Center: (7, -2); Radius: 5; Graph the circle as described.

Step by step solution

01

Identify the Circle Equation Form

The given equation \((x-7)^{2}+(y+2)^{2}=25\) is in the standard form of a circle equation \((x-h)^{2}+(y-k)^{2}=r^{2}\), where \(h,k\) is the center and \(r\) is the radius.
02

Determine the Center

In the equation \((x-7)^{2}+(y+2)^{2}=25\), compare it with the standard form to identify \(h\) and \(k\). Thus, the center \(h\) is 7 and \(k\) is -2. Therefore, the center of the circle is \( (7, -2) \).
03

Determine the Radius

The radius \(r\) is the square root of the constant term on the right side of the equation. In this case, \25 = r^{2}\, so \ r = \sqrt{25} = 5 \.
04

Graph the Circle by Hand

1. Plot the center of the circle at \(7, -2\).\2. Use the radius \(5\) to mark points that are 5 units away from the center in all directions (up, down, left, right, and diagonal).\3. Draw the circle through these points.
05

Verify with a Graphing Calculator

Input the equation \((x-7)^{2}+(y+2)^{2}=25\) into a graphing calculator to confirm the accuracy of the graphed circle.

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Key Concepts

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

Circle Center
In the equation of a circle, the center is represented by the coordinates \( h \) and \( k \). This can be understood using the standard form of a circle equation: \[ (x-h)^{2}+(y-k)^{2}=r^{2} \]. Here, \( h \) and \( k \) denote the center.

For example, in the equation \[ (x-7)^{2}+(y+2)^{2}=25 \], by comparing it to the standard form, you can see that:
  • \( h = 7 \)
  • \( k = -2 \)
Hence, the center of the circle is \( (7, -2) \).

Identifying the center is crucial for plotting the circle accurately. Simply find \( h \) and \( k \) by looking at what values of \( x \) and \( y \) are being squared and adjusted inside the parentheses.
Circle Radius
To find the radius of a circle from its equation, we must identify the value of \( r \). In the standard form \[ (x-h)^{2}+(y-k)^{2}=r^{2} \], \( r \) squared is the constant on the right side of the equation.

Let's take our example \[ (x-7)^{2}+(y+2)^{2}=25 \]. Here:
  • \[ r^{2} = 25 \]
To find \( r \), take the square root:
  • \[ r = \sqrt{25} = 5 \]
Therefore, the radius is \( 5 \).

The radius helps you measure how far all points on the circle are from the center, ensuring you can plot the circle correctly on a graph.
Graphing Circle
Graphing a circle involves a few simple steps once you know the center and the radius. With the equation \[ (x-7)^{2}+(y+2)^{2}=25 \], we have the center \( (7,-2) \) and radius \( 5 \). Follow these steps to graph the circle:

  • Plot the center at \( 7,-2 \) on your graph.
  • From the center, move 5 units in each direction (up, down, left, right, and diagonals).
  • Mark these points to guide your drawing.
  • Connect the points smoothly to form the circle.

There isn't just one way to graph a circle 鈥撯�� you can use graphing calculators too. Simply input the equation to visualize it. For example, input \[ (x-7)^{2}+(y+2)^{2}=25 \] into your graphing calculator to check your hand-drawn graph鈥檚 accuracy. With these steps, you will effectively understand and graph circles with ease.

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Most popular questions from this chapter

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