Worksheet: Equation of a Circle Passing through Three Noncollinear Points

In this worksheet, we will practice finding the equation of a circle passing through three noncollinear points that form a right triangle.

Q1:

Find the equation of the circle that passes through the points 𝐴(4,3), 𝐵(3,4), and 𝐶(2,3).

  • A(𝑥+3)+(𝑦+3)=1
  • B(𝑥3)+(𝑦3)=1
  • C(𝑥+2)+(𝑦+3)=2
  • D(𝑥+6)+(𝑦+6)=2

Q2:

Determine the general equation of the shown circle 𝑀 passing through the origin point and the two points 𝐴(8,0) and 𝐵(0,10).

  • A𝑥+𝑦+8𝑥10𝑦=0
  • B𝑥+𝑦+10𝑥8𝑦=0
  • C𝑥+𝑦16𝑥+20𝑦=0
  • D𝑥+𝑦8𝑥+10𝑦=0

Q3:

Find the general equation of the circle through the origin that also passes through (12,0) and (0,16).

  • A𝑥+𝑦6𝑥8𝑦=0
  • B𝑥+𝑦12𝑥16𝑦=0
  • C𝑥+𝑦+12𝑥+16𝑦=0
  • D𝑥+𝑦24𝑥32𝑦+300=0

Q4:

Find the general form of the equation of a circle that touches the 𝑥-axis and passes through the two points (6,9) and (1,2).

  • A𝑥+𝑦+6𝑥+10𝑦+25=0, 𝑥+𝑦18𝑥+34𝑦+289=0
  • B𝑥+𝑦+6𝑥+10𝑦+6=0, 𝑥+𝑦18𝑥+34𝑦+81=0
  • C𝑥+𝑦+6𝑥+10𝑦+9=0, 𝑥+𝑦18𝑥+34𝑦+81=0
  • D𝑥+𝑦+3𝑥+5𝑦+9=0, 𝑥+𝑦18𝑥+34𝑦+81=0

Q5:

Find the centre of the circle through points 𝐴(3,1), 𝐵(1,2), and 𝐶(1,2).

  • A(1.5,0.5)
  • B(2,0)
  • C(0,1.5)
  • D(1,0.5)

Q6:

The points 𝐴(1,1), 𝐵(1,5), 𝐶(17,11), and 𝐷(19,5) form a rectangle. What is the equation of the circle that contains all four points?

  • A(𝑥+9)+(𝑦+5)=400
  • B(𝑥9)+(𝑦+5)=40
  • C(𝑥+9)+(𝑦5)=360
  • D(𝑥9)+(𝑦5)=100

Q7:

The coordinates for three of a group of aerialists in a circular formation are 𝐺(23,9), 𝐻(12,2), and 𝐽(12,20). If each unit represents 1 foot, determine the diameter of their circular formation.

Q8:

Find the equation of the circle that passes through the points 𝐴(2,1), 𝐵(5,2), and 𝐶(2,5).

  • A(𝑥2)+(𝑦2)=9
  • B(𝑥+2)+(𝑦+2)=9
  • C(𝑥2)+(𝑦5)=18
  • D(𝑥4)+(𝑦4)=6

Q9:

Find the equation of the circle that passes through the points 𝐴(6,1), 𝐵(3,10), and 𝐶(12,1).

  • A(𝑥+3)+(𝑦1)=81
  • B(𝑥3)+(𝑦+1)=81
  • C(𝑥+12)+(𝑦1)=162
  • D(𝑥+6)+(𝑦2)=18

Q10:

Find the equation of the circle that passes through the points 𝐴(8,7), 𝐵(1,8), and 𝐶(0,1).

  • A(𝑥4)+(𝑦4)=25
  • B(𝑥+4)+(𝑦+4)=25
  • C𝑥+(𝑦1)=50
  • D(𝑥8)+(𝑦8)=10

Q11:

Find the equation of the circle that passes through the points 𝐴(1,6), 𝐵(0,1), and 𝐶(7,0).

  • A(𝑥+3)+(𝑦+3)=25
  • B(𝑥3)+(𝑦3)=25
  • C𝑦+(𝑥+7)=50
  • D(𝑥+6)+(𝑦+6)=10

Q12:

Find the equation of the circle that passes through the points 𝐴(12,2), 𝐵(4,6), and 𝐶(4,2).

  • A(𝑥4)+(𝑦+2)=64
  • B(𝑥+4)+(𝑦2)=64
  • C(𝑥+4)+(𝑦+2)=128
  • D(𝑥8)+(𝑦+4)=16

Q13:

Find the equation of the circle that passes through the points 𝐴(5,4), 𝐵(2,5), and 𝐶(3,2).

  • A(𝑥1)+(𝑦1)=25
  • B(𝑥+1)+(𝑦+1)=25
  • C(𝑥+3)+(𝑦+2)=50
  • D(𝑥2)+(𝑦2)=10

Q14:

Find the equation of the circle that passes through the points 𝐴(2,3), 𝐵(3,8), and 𝐶(8,3).

  • A(𝑥+3)+(𝑦3)=25
  • B(𝑥3)+(𝑦+3)=25
  • C(𝑥+8)+(𝑦3)=50
  • D(𝑥+6)+(𝑦6)=10

Q15:

Find the equation of the circle that passes through the points 𝐴(4,2), 𝐵(6,4), and 𝐶(4,6).

  • A(𝑥+4)+(𝑦+4)=4
  • B(𝑥4)+(𝑦4)=4
  • C(𝑥+4)+(𝑦+6)=8
  • D(𝑥+8)+(𝑦+8)=4

Q16:

How many circles can pass through three vertices of a parallelogram?

Q17:

Find the general equation of the circle 𝑀 if the circle touches the 𝑥-axis at 𝐴(8,0) and intersects the 𝑦-axis at 𝐵 and 𝐶(0,16).

  • A𝑥+𝑦+8𝑥+10𝑦+64=0
  • B𝑥+𝑦16𝑥20𝑦+64=0
  • C𝑥+𝑦16𝑥20𝑦+8=0
  • D𝑥+𝑦+8𝑥+16𝑦+64=0

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