Lesson Identify elements of a circle given by its general equation

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Identify elements of a circle given by its general equation

This lesson shows you by examples how to identify elements of a circle given by its general equation.
The method is to transform the given general equation to the standard form using completing the squares.

Problem 1

Identify the center and the radius of a circle given by a general equation x%5E2+%2B+y%5E2+-+6x+%2B+10y+-+15

.

Solution

Apply completing the squares method to transform the given general equation of the circle to its standard form.

Make completing the squares separately for x-group terms and for y-group terms.


 =    --->   (move the constant term to the right, if necessary)  --->

 =      --->   (group the terms containing x and y separately)  ---> 

 =   --->   (Add constant terms to x-group and y-group to complete the squares)

 = ,   (add the same constant terms to the right side to keep the balance)  --->

 
 
 = ,

 = .


Thus you got the equation of the circle in the standard form.       

The center of the circle is the point  (3,-5).

The radius of the circle is r = 7 units.



 
  


Circle  +  = 49

Problem 2

Identify the center and the radius of a circle given by a general equation 2x%5E2+%2B+2y%5E2+-+8x+%2B+4y+-+14

.

Solution

 =   --->  (Divide both sides by 2 to get the leading coefficient at  and  equal to 1)  --->


Apply completing the squares method to transform the given general equation of the circle to its standard form.

Make completing the squares separately for x-group terms and for y-group terms.


 =     --->   (move the constant term to the right, if necessary)  --->

 =       --->   (group the terms containing x and y separately)  ---> 

 =    --->   (Add constant terms to x-group and y-group to complete the squares)

 = ,   (add the same constant terms to the right side to keep the balance)  --->

 
 
 = ,

 = .


Thus you got the equation of the circle in the standard form.       

The center of the circle is the point  (2,-1).

The radius of the circle is r =  =  units.


 
  


Circle  +  = 12