Worked examples
The question
Question: write the equation of the circle with center (0, 0) and radius 5.
center (0, 0) and radius 5
1 Draw the circle
Draw the grid.
Put the center on the grid: (0, 0).
The radius is 5: go 5 from the center.
Every point 5 away from the center makes the circle.
2 Across and up
Take any point (x, y) on the circle.
Go across: from x = 0 to x is x − 0.
Go up: from y = 0 to y is y − 0.
Across and up make a right angle.
The radius 5 is the longest side.
3 Use a² + b² = c²
The rule for every right triangle: a² + b² = c².
a2 + b2 = c2
Put in a, b and c: (x − 0)² + (y − 0)² = 5².
(x − 0)2 + (y − 0)2 = 52
Taking away 0 changes nothing.
− 0: nothing changes
So it becomes x² + y² = 5².
x2 + y2 = 52
5² means 5 times 5.
That is 25.
52 = 5 × 5 = 25
So the equation is x² + y² = 25.
x2 + y2 = 25
This works for every circle with center (h, k) and radius r.
(x − h)2 + (y − k)2 = r2
4 Check
Let's check with a point on the circle: (5, 0).
Put x = 5 and y = 0 in.
We get 5² + 0².
52 + 02 = 52 + 02
25 plus 0 is 25, the number on the right.
Right!
= 25 + 0 = 25
Let's look back.
First: draw the circle.
Then: across and up.
Then: use a² + b² = c².
Last: check.
▶ Watch it solved
The question
Question: write the equation of the circle with center (2, −3) and radius 4.
center (2, −3) and radius 4
1 Draw the circle
Draw the grid.
Put the center on the grid: (2, −3).
The radius is 4: go 4 from the center.
Every point 4 away from the center makes the circle.
2 Across and up
Take any point (x, y) on the circle.
Go across: from x = 2 to x is x − 2.
Go up: from y = −3 to y is y − (−3).
Across and up make a right angle.
The radius 4 is the longest side.
3 Use a² + b² = c²
The rule for every right triangle: a² + b² = c².
a2 + b2 = c2
Put in a, b and c: (x − 2)² + (y − (−3))² = 4².
(x − 2)2 + (y − (−3))2 = 42
Taking away a negative is the same as adding.
− (−) = +
So it becomes (x − 2)² + (y + 3)² = 4².
(x − 2)2 + (y + 3)2 = 42
4² means 4 times 4.
That is 16.
42 = 4 × 4 = 16
So the equation is (x − 2)² + (y + 3)² = 16.
(x − 2)2 + (y + 3)2 = 16
Careful: the signs turn.
Center 2 gives x − 2.
center (2, −3) → x − 2, y + 3
This works for every circle with center (h, k) and radius r.
(x − h)2 + (y − k)2 = r2
4 Check
Let's check with a point on the circle: (6, −3).
Put x = 6 and y = −3 in.
We get 4² + 0².
(6 − 2)2 + (−3 + 3)2 = 42 + 02
16 plus 0 is 16, the number on the right.
Right!
= 16 + 0 = 16
Let's look back.
First: draw the circle.
Then: across and up.
Then: use a² + b² = c².
Last: check.
Answer (x − 2)2 + (y + 3)2 = 16
▶ Watch it solved
The question
Question: find the center and the radius of x² + (y − 5)² = 20.
x2 + (y − 5)2 = 20
1 Compare with the rule
Every circle has this form: (x − h)² + (y − k)² = r².
The center is (h, k).
The radius is r.
(x − h)2 + (y − k)2 = r2
2 Find the center
x² is the same as (x − 0)²: taking away 0 changes nothing.
x2 = (x − 0)2
So h = 0.
h = 0
y − 5 is like y − k.
So k = 5.
k = 5
y − 5 ↔ y − k
So the center is (0, 5).
center = (0, 5)
3 Find the radius
r² is the number on the right: r² = 20.
r2 = 20
r times r is 20.
So r is the square root of 20.
r = √20
20 is not a square number.
So we keep √20.
That is the exact answer.
4 × 4 = 16, 5 × 5 = 25
With a calculator, √20 is about 4.5, to the nearest tenth.
√20 ≈ 4.5
So the center is (0, 5) and the radius is √20.
center (0, 5) and r = √20
4 Check
Let's check: draw the circle.
Center (0, 5), radius about 4.5.
Put the center on the grid.
Draw the circle around it.
This point is on the circle: (2, 9).
It is 2 across and 4 up from the center.
Put x = 2 and y = 9 in.
We get 2² + 4².
22 + (9 − 5)2 = 22 + 42
4 plus 16 is 20, the number on the right.
Right!
= 4 + 16 = 20
Let's look back.
First: compare with the rule.
Then: find the center.
Then: find the radius.
Last: check.
Answer center (0, 5) and r = √20
▶ Watch it solved
