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Chapter 0: Problem 11

Find the equation of the circle satisfying the given conditions. Center \((1,1),\) radius 1

Short Answer

Expert verified
The equation of the circle is \((x - 1)^2 + (y - 1)^2 = 1.\)

Step by step solution

01

Recall the Standard Equation of a Circle

The standard equation of a circle with center \((h, k)\) and radius \(r\) is given by the formula: \[(x - h)^2 + (y - k)^2 = r^2.\]
02

Identify the Given Information

From the problem statement, the center of the circle \((h, k)\) is given as \((1, 1)\), and the radius \(r\) is 1.
03

Substitute the Given Values into the Equation

Replace \(h\), \(k\), and \(r\) in the standard circle equation \((x - h)^2 + (y - k)^2 = r^2\) with the values from our problem: \((x - 1)^2 + (y - 1)^2 = 1^2.\)
04

Simplify the Equation

Simplify the equation obtained in step 3:\[(x - 1)^2 + (y - 1)^2 = 1.\]

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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
To understand the equation of a circle, let's start with the standard form. This equation helps you describe any circle on a coordinate plane. It is given by the formula: \[ (x - h)^2 + (y - k)^2 = r^2 \]. What does this mean exactly? Well, in the equation:
  • \((x - h)^2\) and \((y - k)^2\) are the expressions that adjust the position of the circle on the plane.
  • \((h, k)\) is the center of the circle.
  • \(r\) stands for radius, which is the distance from the center to any point on the circle.

This form is useful because it directly shows the circle's center and radius. By substituting the specific values for a circle's center and radius into this formula, you get the precise equation defining that circle.
Radius
The radius of a circle is a fundamental concept you need to understand in geometry. Simply put, it is the distance from the center of the circle to any point on the circle itself. In our standard equation of a circle, the radius is represented by the letter \(r\) as in:\[ r^2 \].Why is the radius important? Here's why:
  • The length of the radius tells you how big or small the circle is.
  • A larger radius means a bigger circle, whereas a smaller radius results in a smaller circle.
  • The radius remains constant for any given circle, regardless of which direction you measure.

By knowing the radius and the center, you can easily draw the circle using the standard equation \((x - h)^2 + (y - k)^2 = r^2\).
Center of a Circle
The center of a circle is as important as its radius. It designates the exact point around which the circle is perfectly symmetrical. In the standard equation form, the center is denoted by the coordinates \((h, k)\). These values specifically determine the exact location of a circle on a coordinate plane. Consider this:
  • Moving the center changes the position of the circle entirely but not its size or shape.
  • The circle remains round and identical in size regardless of where its center is located.
  • The center point is the middle point from which every edge of the circle extends equally, defined as the radius.

By inserting the center values \(h\) and \(k\) into the standard form \((x - h)^2 + (y - k)^2 = r^2\), you get the equation that represents the circle's position on the graph.

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Most popular questions from this chapter

Sketch the graph of \(f(x)=(x-2)^{2}-4\) using translations

Use a computer or a graphing calculator in Problems \(37-40 .\) Let \(f(x)=2 \sqrt{x}-2 x+0.25 x^{2}\). Using the same axes, draw the graphs of \(y=f(x), y=f(1.5 x),\) and \(y=\) \(f(x-1)+0.5,\) all on the domain [0,5]

Plot the graph of each equation. Begin by checking for symmetries and be sure to find all \(x\) - and \(y\) -intercepts. \(x^{4}+y^{4}=16\)

Find the points of intersection of the graphs of \(y=x^{2}-2 x+4\) and \(y-x=4\)

Plot the graphs of both equations on the same coordinate plane. Find and label the points of intersection of the two graphs (see Example 4). $$ \begin{array}{l} y=-2 x+3 \\ y=-2(x-4)^{2} \end{array} $$
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