Circle: Equation
How to find and use the equation of a circle: formula, 4 examples, and their solutions.
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.
ExampleCenter: (2, 1), r = 3
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.
ExampleCenter: (4, 0), Diameter: 14
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.
ExampleEndpoints: (-1, 3), (7, 1)
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.
ExampleGeneral Form → Center, Radius
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.