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Conic Sections: Circles
In this lesson you will learn how to write equations of circles and graphs of circles will be compared to their equations.

Definition: A circle is all points equidistant (the distance is called the radius) from one point (which is called the center of the circle). A circle can be formed by slicing a right circular cone with a plane traveling parallel to the base of the cone. This effect can be seen in the following video and screen captures.



Part I. The graph of a circle with radius 3 and center at the origin is shown below.

Note that it is sometimes not clear that the top half and bottom half of the circle are connected because of the way a graphing calculator draws the top and bottom halves separately.

A graph using other graphing software, such as EXCEL, can present the equation more accurately. The graph shown below was produced using EXCEL's chart feature.

An equation of this circle can be found by using the distance formula. We calculate the distance from the point on the circle (x, y) to the origin (0, 0). This distance is the radius, which is 3 in this example:

.
Squaring both sides and simplifying, we have
.

Examples
Example Using the model discussed above, what is the equation of a circle that has its center at the origin and a radius of 2?
What is your answer?
 
Example
What is the radius of the circle ? Describe what its graph would look like.
What is your answer?
 
Example
What is the radius of the circle ? Describe what its graph would look like.
What is your answer?
 

Part II. Now, suppose the center is not at the origin (0, 0) but is at some other point such as (2, -1). Graphs generated from both a graphing calculator graph and a spreadsheet are shown below. In each case, the radius is 3.
Again we will use the distance formula to derive the equation of the circle. Our two points will be (x,y) - a general point on the circle; and (2, -1) - the center of the circle. The radius (or distance between the two points) will be 3.

.

Squaring and simplifying, we have
.

Examples
Example Use the second model discussed above to determine the equation of a circle that has its center at (2, -1) and a radius of 2?
What is your answer?
 
Example What would be the equation of a circle that has its center at (2, -1) and a radius of ?
What is your answer?
 

Part III. Since the equation of a circle is written by using the distance formula from (x, y) to the center and setting this equal to the radius, we are now ready to build a standard formula for a circle.
Let the center be the point (h, k) and the radius be r.

The distance formula gives
.
Squaring both sides gives the standard equation of a circle:
.

Examples
Example
Use the standard equation form to determine the equation of the circle that has a center at (2, 3) and a radius of 4? Then describe what the graph would look like.
What is your answer?
 
Example
Use the standard equation form to determine the equation of the circle that has a center at (-3, -3) and a radius of 2? Then describe what the graph would look like.
What is your answer?
 
Example
Using the standard equation form discussed above, find the equation and graph of a circle with: center (0,-1) and radius
What is your answer?
 

Part IV. Writing an equation for a circle in standard form and getting a graph sometimes involves some algebra.
For example, the equation is an equation of a circle. To see this we will need to complete the square for both x and y.

.
This simplifies to
which is the standard form of a circle with center (2, -3) and radius = 6.

To graph a circle in standard form, you need to first solve for y. What follows are two examples showing you the steps you need to follow in using your graphing calculator to produce a graph of a circle.
  • Given the equation
  • then
    Enter this into the calculator as
    .
    The calculator will graph the top and bottom halves of the circle using Y1 and Y2.
  • Given the equation
  • ,
    then
    .
    Enter this into the calculator as
    .
    The calculator will graph the top and bottom halves of the circle using Y1 and Y2. In addition to the graph generated by the graphing calculator, a graph produced by EXCEL is also presented. Both graphs have their centers at (-2,3) and radii = 6.

Examples
Put each of the following equations into standard form and then graph each one. Be sure that you can then identify the coordinates of its center and the length of its radius.
Example
What is your answer?
 
Example
What is your answer?
 
Example
What is your answer?
 
Example
What is your answer?
 



M Ransom
J Anderson

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