Circles - Find the Radius and Center - by Completing the Square
Circles - Find the Radius and the Center - How it Works - Video
Example 1
Example 1:
In this examples when we don't have parentheses, we have to complete the square to find our center and radius.
Our first step to arrange our terms so that x's and y's are next to each other and the constant is on the other side of the equal sign.
x2 + y2 - 4x + 6y - 36 = 0
x2 - 4x + y2 + 6y = 36
(x2 - 4x + __ ) + (y2 + 6y + __ ) = 36 + __ + __
(x2 - 4x + _4_ ) + (y2 + 6y + _9_ ) = 36 + _4_ + _9_
(x - 2)2 + (y + 3)2 = 49
Rearranged the terms.
Added empty spaces to complete the square.
Divided b by 2 for both x and y and squared. Added on both sides.
Factored and added constant terms.
Using the equation, (x - 2)2 + (y + 3)2 = 49, we can find the center and the radius.
Remember the standard equation is (x - h)2 + (y - k)2 = r2. Our center has moved 2 units horizontally and 3 units vertically. Now we have to decide whether each number is negative or positive. What number do we substitute in for (x - 2)2 to get back to center? What number do we substitute in for (y + 3)2 to get back to center? So our center is (2, -3).
As for our radius, we square root 49 and the result is 7. So our radius is 7.
Example 2
Example 2:
In this examples when we don't have parentheses, we have to complete the square to find our center and radius.
Our first step to arrange our terms so that x's and y's are next to each other and the constant is on the other side of the equal sign.
x2 + y2 + 4y - 117 = 0
x2 + y2 + 4y = 117
(x2 ) + (y2 + 4y + __ ) = 117 + __
(x2 ) + (y2 + 4y + _4_ ) = 117 + _4_
(x - 0)2 + (y + 2)2 = 121
Rearranged the terms.
Added empty spaces to complete the square.
Divided b by 2 for y and squared. Added on both sides.
Factored and added constant terms.
Using the equation, (x - 0)2 + (y + 2)2 = 121, we can find the center and the radius. Now you usually don't see the 0 in the first parenthesis, but it will help us find the center.
Remember the standard equation is (x - h)2 + (y - k)2 = r2. Our center has moved 0 units horizontally and 2 units vertically. Now we have to decide whether each number is negative or positive. What number do we substitute in for (x - 0)2 to get back to center? What number do we substitute in for (y + 2)2 to get back to center? So our center is (0, -2).
As for our radius, we square root 121 and the result is 11. So our radius is 11.
Example 3
Example 3:
In this examples when we don't have parentheses, we have to complete the square to find our center and radius.
Our first step to arrange our terms so that x's and y's are next to each other and the constant is on the other side of the equal sign. Before we do that, let's take a look at the all the numbers. Each one is a multiple of 2 so let's divide each number by 2 to make our numbers smaller. This is also helps us out because it makes completing the square more complicated.
2x2 + 2y2 - 12x + 16y - 22 = 0
x2 + y2 - 6x + 8y - 11 = 0
x2 - 6x + y2 + 8y = 11
(x2 - 6x + __ ) + (y2 + 8y + __ ) = 11 + __ + __
(x2 - 6x + _9_ ) + (y2 + 8y + _16_ ) = 11 + _9_ + _16_
(x - 3)2 + (y + 4)2 = 36
Divided each number 2.
Rearranged the terms.
Added empty spaces to complete the square.
Divided b by 2 for both x and y and squared. Added on both sides.
Factored and added constant terms.
Using the equation, (x - 3)2 + (y + 4)2 = 36, we can find the center and the radius.
Remember the standard equation is (x - h)2 + (y - k)2 = r2. Our center has moved 3 units horizontally and 4 units vertically. Now we have to decide whether each number is negative or positive. What number do we substitute in for (x - 3)2 to get back to center? What number do we substitute in for (y + 4)2 to get back to center? So our center is (3, -4).
As for our radius, we square root 36 and the result is 6. So our radius is 6.