Circle: Equation

How to find and use the equation of a circle: formula, 4 examples, and their solutions.

Formula

Formula

If the center of a circle is (h, k)
and if the radius is r,

then the equation of the circle is
(x - h)2 + (y - k)2 = r2.

Example 1

Example

Solution

The center of the circle is (2, 1).
r = 3

Then the equation of the circle is
(x - 2)2 + (y - 1)2 = 32.

32 = 9

So
(x - 2)2 + (y - 1)2 = 9
is the answer.

Example 2

Example

Solution

The center of the circle is (4, 0).

The diameter of the circle is 14.

So the radius r is, 14/2, 7.

Center: (4, 0)
r = 7

Then the equation of the circle is
(x - 4)2 + (y - 0)2 = 72.

+(y - 0)2 = +y2

72 = 49

So
(x - 4)2 + y2 = 49
is the answer.

Example 3

Example

Solution

First draw the condition.

Draw a circle.
Draw the diameter.
And draw the endpoints of the diameter
(-1, 3) and (7, 1).

The midpoint of the diameter M
is the center of the circle.

And the divided segments
are the r (radius).

Find the center M.

M is the midpoint of (-1, 3) and (7, 1).

So M([-1 + 7]/2, [3 + 1]/2).

-1 + 7 = 6
3 + 1 = 4

6/2 = 3
4/2 = 2

So M(3, 2).

Write (3, 2)
on the center of the circle.

Find the radius r.

r is the distance between
the center M(3, 2) and the endpoint (7, 1).

So r = √(7 - 3)2 + (1 - 2)2

Distance Formula

You can also choose
M(3, 2) and the other endpoint (-1, 3).
You'll get the same answer.

7 - 3 = 4
1 - 2 = -1

42 = 16
+(-1)2 = +1

16 + 1 = 17

So r = √17.

Write √17
on the radius r of the circle.

Center: (3, 2)
r = √17

Then the equation of the circle is
(x - 3)2 + (y - 2)2 = (√17)2.

(√17)2 = 17

Square Root

So
(x - 3)2 + (y - 2)2 = 17
is the answer.

Example 4

Example

Solution

The given circle is in general form:
x2 + y2 + Ax + By + C = 0.

To find the center and the r,
change this equation to standard form:
(x - h)2 + (y - k)2 = r2.

First, move the constant term +20
to the right side.

Use x2 - 4x
to make a perfect square trinomial.

x2 is x2.

-4x is
-2 times
x times,
(-4x)/(-2⋅x), 2.

Write +22.

Quadratic Equation: Completing the Square

Use +y2 + 10y
to make a perfect square trinomial.

y2 is y2.

+10y is
+2 times
y times,
(+10y)/(+2⋅y), 5.

Write +52.

Write the right side -20.

And to undo +22 and +52,
write +22 + 52
on the right side.

x2 - 2⋅x⋅2 + 22
= (x - 2)2

y2 + 2⋅y⋅5 + 52
= (y + 5)2

Factor a Perfect Square Trinomial

+22 = +4
+52 = +25

-20 + 25 = 5

5 + 4 = 9

To find the radius easily,
change 9 to 32.

To find the y value of the center easily,
change +(y + 5)2 to +(y - (-5))2.

Then (x - 2)2 + (y - (-5))2 = 32.

See the circle equation
(x - 2)2 + (y - (-5))2 = 32.

The center is (2, -5).
The radius is 3.

So
Center: (2, -5)
Radius: 3
is the answer.