Circles: equation, center and radius

About this lesson

Geometry › Coordinate geometry

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Circles: equation, center and radius

Warm-up

Three quick questions from earlier lessons.

Discover

A point (x,y)(x, y) is on the circle with center (2,1)(2, 1) and radius 3 when its distance to (2,1)(2, 1) is 3. Which equation says that?

  1. (x−2)2+(y−1)2=9(x - 2)^2 + (y - 1)^2 = 9
  2. (x+2)2+(y+1)2=9(x + 2)^2 + (y + 1)^2 = 9
  3. (x−2)2+(y−1)2=3(x - 2)^2 + (y - 1)^2 = 3

Key idea

A circle is the set of points at a fixed distance rr from a center (h,k)(h, k).

(x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2. From the expanded form x2+y2+Dx+Ey+F=0x^2 + y^2 + Dx + Ey + F = 0, complete the square in xx and in yy to read the center and radius.

Worked example

Find the center and radius of x2+y2−4x+6y−3=0x^2 + y^2 - 4x + 6y - 3 = 0.

(x−2)2−4+(y+3)2−9−3=0(x - 2)^2 - 4 + (y + 3)^2 - 9 - 3 = 0
Group and complete the squares: x² − 4x = (x − 2)² − 4, y² + 6y = (y + 3)² − 9.
(x−2)2+(y+3)2=16(x - 2)^2 + (y + 3)^2 = 16
Move the numbers to the right.
Center (2,−3)(2, -3), radius 44
Read h, k and r = √16.

Your turn, guided

Your turn: (x+5)2+(y−1)2=49(x + 5)^2 + (y - 1)^2 = 49.

Center (−5,1)(-5, 1)
x + 5 = x − (−5).
Radius 77
√49.

Practice ★

  1. What is the radius of the circle (x−9)2+(y+1)2=16(x - 9)^2 + (y + 1)^2 = 16?
  2. What is the radius of the circle (x−1)2+(y+7)2=64(x - 1)^2 + (y + 7)^2 = 64?
  3. What is the radius of the circle (x−4)2+(y+9)2=16(x - 4)^2 + (y + 9)^2 = 16?

Watch out!

Practice ★★

  1. Find the center and radius of the circle x2+y2−10x+4y+25=0x^2 + y^2 - 10x + 4y + 25 = 0.
  2. Find the center and radius of the circle x2+y2−2x+4y+−11=0x^2 + y^2 - 2x + 4y + -11 = 0.
  1. Which is the equation of the circle with center (2,−8)(2, -8) and radius 7?
    1. (x−2)2+(y+8)2=49(x - 2)^2 + (y + 8)^2 = 49
    2. (x+2)2+(y−8)2=49(x + 2)^2 + (y - 8)^2 = 49
    3. (x−2)2+(y+8)2=7(x - 2)^2 + (y + 8)^2 = 7
  2. Which is the equation of the circle with center (7,−2)(7, -2) and radius 9?
    1. (x−7)2+(y+2)2=81(x - 7)^2 + (y + 2)^2 = 81
    2. (x+7)2+(y−2)2=81(x + 7)^2 + (y - 2)^2 = 81
    3. (x−7)2+(y+2)2=9(x - 7)^2 + (y + 2)^2 = 9

Exit ticket

Three questions to check you have got it.

  1. Which is the equation of the circle with center (2,−8)(2, -8) and radius 7?
    1. (x−2)2+(y+8)2=49(x - 2)^2 + (y + 8)^2 = 49
    2. (x+2)2+(y−8)2=49(x + 2)^2 + (y - 8)^2 = 49
    3. (x−2)2+(y+8)2=7(x - 2)^2 + (y + 8)^2 = 7
  2. Which is the equation of the circle with center (7,−2)(7, -2) and radius 9?
    1. (x−7)2+(y+2)2=81(x - 7)^2 + (y + 2)^2 = 81
    2. (x+7)2+(y−2)2=81(x + 7)^2 + (y - 2)^2 = 81
    3. (x−7)2+(y+2)2=9(x - 7)^2 + (y + 2)^2 = 9
  3. Which is the equation of the circle with center (6,−1)(6, -1) and radius 6?
    1. (x−6)2+(y+1)2=36(x - 6)^2 + (y + 1)^2 = 36
    2. (x+6)2+(y−1)2=36(x + 6)^2 + (y - 1)^2 = 36
    3. (x−6)2+(y+1)2=6(x - 6)^2 + (y + 1)^2 = 6

Summary card

Circles

  • (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2: center (h,k)(h, k), radius rr.
  • Expanded form: complete the square in xx and in yy.

Watch out: (x+3)(x + 3) means h=−3h = -3, and the right side is r2r^2.

End of the lesson

Next lesson: Parabolas: focus and directrix Back to the chapter

Answers: Circles: equation, center and radius

A

  1. 44
  2. 88
  3. 44

B

  1. 22
  2. 44

C

  1. (x−2)2+(y+8)2=49(x - 2)^2 + (y + 8)^2 = 49
  2. (x−7)2+(y+2)2=81(x - 7)^2 + (y + 2)^2 = 81

D

  1. (x−2)2+(y+8)2=49(x - 2)^2 + (y + 8)^2 = 49
  2. (x−7)2+(y+2)2=81(x - 7)^2 + (y + 2)^2 = 81
  3. (x−6)2+(y+1)2=36(x - 6)^2 + (y + 1)^2 = 36

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