Worked examples
The question
Question: find the center and the radius of x² + y² + 8y = 0.
x2 + y2 + 8y = 0
1 Group x and y
Put the x terms together in a bracket, and the y terms together.
x2 + (y2 + 8y) = 0
2 Complete the square for y
To make y² + 8y a full square: half of 8 is 4.
4 times 4 is 16.
So we add 16.
We do it on both sides, so they stay equal.
x2 + (y2 + 8y + 16) = 0 + 16
8 ÷ 2 = 4, 4 × 4 = 16
Careful!
What we add on the left, we add on the right too.
Then the two sides stay equal.
Careful: add 16 on both sides
3 Write as squares
Each bracket is now a full square.
On the right, 0 + 16 is 16.
So x² + (y + 4)² = 16.
x2 + (y + 4)2 = 16
y² + 8y + 16 is (y + 4)².
y² + 8y + 16 = (y + 4)2
4 Read the center and radius
Compare with the circle rule: (x − h)² + (y − k)² = r².
(x − h)2 + (y − k)2 = r2
The signs turn: x² gives h = 0.
y + 4 gives k = −4.
y + 4 → k = −4
x → h = 0
So the center is (0, −4).
center = (0, −4)
r² is 16.
So r is the square root of 16: 4, because 4 times 4 is 16.
r = √16 = 4
So the center is (0, −4) and the radius is 4.
center (0, −4) and r = 4
5 Check
Let's check with a point on the circle: (4, −4), 4 to the right from the center.
Put it in the first equation.
4² + (−4)² + 8 × (−4) = 16 + 16 − 32 = 0
We get 0, the same as the right side.
So the center and the radius are right!
Let's look back.
First: group x and y.
Then: complete the square for y.
Then: write as squares.
Then: read the center and radius.
Last: check.
Answer center (0, −4) and r = 4
▶ Watch it solved
The question
Question: find the center and the radius of x² + y² − 4x + 6y − 3 = 0.
x2 + y2 − 4x + 6y − 3 = 0
1 Group x and y
Put the x terms together in a bracket, and the y terms together.
(x2 − 4x) + (y2 + 6y) − 3 = 0
2 Remove the − 3
We want only the x and y groups on the left.
To remove − 3, we add 3.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
(x2 − 4x) + (y2 + 6y) − 3 + 3 = 0 + 3
− 3 and + 3 make zero.
Cross them out.
0 plus 3 is 3.
(x2 − 4x) + (y2 + 6y) = 3
3 Complete the square for x
To make x² − 4x a full square: half of −4 is −2.
(−2) times (−2) is 4.
So we add 4.
We do it on both sides, so they stay equal.
(x2 − 4x + 4) + (y2 + 6y) = 3 + 4
−4 ÷ 2 = −2, (−2) × (−2) = 4
Careful!
What we add on the left, we add on the right too.
Then the two sides stay equal.
Careful: add 4 on both sides
4 Complete the square for y
To make y² + 6y a full square: half of 6 is 3.
3 times 3 is 9.
So we add 9.
We do it on both sides, so they stay equal.
(x2 − 4x + 4) + (y2 + 6y + 9) = 3 + 4 + 9
6 ÷ 2 = 3, 3 × 3 = 9
5 Write as squares
Each bracket is now a full square.
On the right, 3 + 4 + 9 is 16.
So (x − 2)² + (y + 3)² = 16.
(x − 2)2 + (y + 3)2 = 16
x² − 4x + 4 is (x − 2)².
y² + 6y + 9 is (y + 3)².
x² − 4x + 4 = (x − 2)2
y² + 6y + 9 = (y + 3)2
6 Read the center and radius
Compare with the circle rule: (x − h)² + (y − k)² = r².
(x − h)2 + (y − k)2 = r2
The signs turn: x − 2 gives h = 2.
y + 3 gives k = −3.
x − 2 → h = 2
y + 3 → k = −3
So the center is (2, −3).
center = (2, −3)
r² is 16.
So r is the square root of 16: 4, because 4 times 4 is 16.
r = √16 = 4
So the center is (2, −3) and the radius is 4.
center (2, −3) and r = 4
7 Check
Let's check with a point on the circle: (6, −3), 4 to the right from the center.
Put it in the first equation.
6² + (−3)² − 4 × 6 + 6 × (−3) − 3
= 36 + 9 − 24 − 18 − 3 = 0
We get 0, the same as the right side.
So the center and the radius are right!
Let's look back.
First: group x and y.
Then: remove the − 3.
Then: complete the square for x.
Then: complete the square for y.
Then: write as squares.
Then: read the center and radius.
Last: check.
Answer center (2, −3) and r = 4
▶ Watch it solved
The question
Question: find the center and the radius of x² + y² − 2x − 4y = 0.
x2 + y2 − 2x − 4y = 0
1 Group x and y
Put the x terms together in a bracket, and the y terms together.
(x2 − 2x) + (y2 − 4y) = 0
2 Complete the square for x
To make x² − 2x a full square: half of −2 is −1.
(−1) times (−1) is 1.
So we add 1.
We do it on both sides, so they stay equal.
(x2 − 2x + 1) + (y2 − 4y) = 0 + 1
−2 ÷ 2 = −1, (−1) × (−1) = 1
Careful!
What we add on the left, we add on the right too.
Then the two sides stay equal.
Careful: add 1 on both sides
3 Complete the square for y
To make y² − 4y a full square: half of −4 is −2.
(−2) times (−2) is 4.
So we add 4.
We do it on both sides, so they stay equal.
(x2 − 2x + 1) + (y2 − 4y + 4) = 0 + 1 + 4
−4 ÷ 2 = −2, (−2) × (−2) = 4
4 Write as squares
Each bracket is now a full square.
On the right, 0 + 1 + 4 is 5.
So (x − 1)² + (y − 2)² = 5.
(x − 1)2 + (y − 2)2 = 5
x² − 2x + 1 is (x − 1)².
y² − 4y + 4 is (y − 2)².
x² − 2x + 1 = (x − 1)2
y² − 4y + 4 = (y − 2)2
5 Read the center and radius
Compare with the circle rule: (x − h)² + (y − k)² = r².
(x − h)2 + (y − k)2 = r2
The signs turn: x − 1 gives h = 1.
y − 2 gives k = 2.
x − 1 → h = 1
y − 2 → k = 2
So the center is (1, 2).
center = (1, 2)
r² is 5.
5 is not a square number.
So we keep √5.
That is the exact answer.
r = √5
With a calculator, √5 is about 2.2, to the nearest tenth.
√5 ≈ 2.2
So the center is (1, 2) and the radius is √5.
center (1, 2) and r = √5
6 Check
Let's check with a point on the circle: (2, 4), 1 across and 2 up from the center.
Put it in the first equation.
2² + 4² − 2 × 2 − 4 × 4 = 4 + 16 − 4 − 16
= 0
We get 0, the same as the right side.
So the center and the radius are right!
Let's look back.
First: group x and y.
Then: complete the square for x.
Then: complete the square for y.
Then: write as squares.
Then: read the center and radius.
Last: check.
Answer center (1, 2) and r = √5
▶ Watch it solved
