8.2—Equations of Circles
8.2 Warm Up
The values for a and b below are lengths of the legs of right triangles.
For each set of leg lengths, find the length of the hypotenuse, c.
1
3
1. a = 10 and b = 8
2. a = 3 and b = 9
3. a = and b =
2
4
8.2—Equations of Circles
Objective: Write the equation of a circle and graph a circle.
y
(x, y)
r
How can you find the radius r of any circle with center (h, k)?
How
can you find the radius r of any circle with
Pick any point (x, y) on the circle and use the distance formula.
center (h, k)? Pick any point (x, y) on the circle
r  (x  h)  (y  k)
and
use the distance formula.
Now square both sides and you have the standard equation of a circle!
2
2
Standard Equation of a Circle with radius r and center (h, k):
(x – h)2 + (y – k)2 = r 2
Now
square both sides and you have the
(If the center is the origin, then the standard equation is x + y = r
standard equation of a circle!
2
2
2
)
Standard Equation of a Circle with radius r and center (h, k):
(x – h)2 + (y – k)2 = r 2
If the center is the origin, then the standard equation is
x 2 + y2 = r 2
General Equation of a Circle with radius r and center (h, k):
𝑨𝒙𝟐 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎
𝑥 2 + 𝑦 2 + 𝐶𝑥 + 𝐷𝑦 + 𝐸 = 0
Consider ʘS , placed on the coordinate plane as shown.
a. What are the coordinates of the center, S?
b. How can you describe the coordinates of R as it
moves around the circle?
c. What is the distance from S to R?
d. Use the distance formula to write a relationship between
SR and the radius of the circle.
Simplify your answer to eliminate any square root symbols.
ʘA is centered at (1,2) and contains point B(4,2). Determine the radius of
ʘA . Let C(x,y) be any other point on ʘA . Using the answer choices
provided, fill in the blanks to write an equation for ʘA .
a. By the definition of a circle and the distance between two points on
a horizontal line, r = AB = _________________.
b. By the definition of a circle and the distance formula, r = AC = ______________.
c. Therefore, 3 = ________________.
d. Squaring both sides and applying the Symmetric Property gives
______________________.
Apply the same reasoning you used on the preceding page to find an equation
for any circle in terms of its center and radius. Consider ʘA with center at point
A(h,k) in the coordinate plane and radius of fixed but unknown length, r. The
distance between two points is 𝒅 = (𝒙𝟐 − 𝒙𝟏 )𝟐 +(𝒚𝟐 − 𝒚𝟏 )𝟐 . Let d = r, the
radius of ʘA . Fill in the blanks using the answer choices provided.
a. (x1 – x2) can be replaced by
___________________.
b. (y1 – y2) = __________________.
c. Substitute into the distance formula
________________________ and square both sides to
get __________________________.
Using the answer choices provided and the standard form for the equation of a circle
that you written above, write an equation for each circle.
ʘA ___________________________
ʘB ___________________________
ʘC ___________________________
ʘD ___________________________
ʘE ___________________________
The standard form for the equation of the circle shown is
(x − 1)2 + (y − 2)2 = 9
Expand each binomial, then simplify the equation and set it equal to
zero to write the equation of this circle in its general form.
(x − 1)2 + (y − 2)2 = 9
x2 − 2x + 1 + y2 − 4y + 4 = 9
x2 − 2x + y2 − 4y − 4 = 0
x2 + y2 − 2x − 4y − 4 = 0
Examples:
Give the center and radius of the circle.
1. (x – 4)2 + (y + 3)2 = 49
2. x2 + (y – 3)2 = ¼
Give the coordinates of the center, the radius, and the equation of the circle.
3.
4.
Write the equation of the circle with the given center and radius in standard and
general form.
5. center (2, –3), radius 8
6. center (0, 5), radius 6
Use the given information to write the standard equation of the circle.
7. The center is (–5, 6),
8. The center is (–2, 0),
a point on the circle is (–1, 3).
the diameter is 6.
Graph the equation.
9. x2 + y2 = 16
10. (x – 1)2 + (y + 2)2 = 9