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 ( 𝑥 + 3 ) + ( 𝑦 + 3 ) = 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).

  • A 𝑥 + 𝑦 1 6 𝑥 + 2 0 𝑦 = 0
  • B 𝑥 + 𝑦 8 𝑥 + 1 0 𝑦 = 0
  • C 𝑥 + 𝑦 + 1 0 𝑥 8 𝑦 = 0
  • D 𝑥 + 𝑦 + 8 𝑥 1 0 𝑦 = 0

Q3:

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

  • A 𝑥 + 𝑦 6 𝑥 8 𝑦 = 0
  • B 𝑥 + 𝑦 + 1 2 𝑥 + 1 6 𝑦 = 0
  • C 𝑥 + 𝑦 2 4 𝑥 3 2 𝑦 + 3 0 0 = 0
  • D 𝑥 + 𝑦 1 2 𝑥 1 6 𝑦 = 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).

  • A 𝑥 + 𝑦 + 6 𝑥 + 1 0 𝑦 + 6 = 0 , 𝑥 + 𝑦 1 8 𝑥 + 3 4 𝑦 + 8 1 = 0
  • B 𝑥 + 𝑦 + 6 𝑥 + 1 0 𝑦 + 2 5 = 0 , 𝑥 + 𝑦 1 8 𝑥 + 3 4 𝑦 + 2 8 9 = 0
  • C 𝑥 + 𝑦 + 3 𝑥 + 5 𝑦 + 9 = 0 , 𝑥 + 𝑦 1 8 𝑥 + 3 4 𝑦 + 8 1 = 0
  • D 𝑥 + 𝑦 + 6 𝑥 + 1 0 𝑦 + 9 = 0 , 𝑥 + 𝑦 1 8 𝑥 + 3 4 𝑦 + 8 1 = 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 ( 1 , 0 . 5 )
  • C ( 2 , 0 )
  • D ( 0 , 1 . 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 ) = 4 0 0
  • B ( 𝑥 + 9 ) + ( 𝑦 5 ) = 3 6 0
  • C ( 𝑥 9 ) + ( 𝑦 + 5 ) = 4 0
  • D ( 𝑥 9 ) + ( 𝑦 5 ) = 1 0 0

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 ( 𝑥 4 ) + ( 𝑦 4 ) = 6
  • B ( 𝑥 2 ) + ( 𝑦 5 ) = 1 8
  • C ( 𝑥 + 2 ) + ( 𝑦 + 2 ) = 9
  • D ( 𝑥 2 ) + ( 𝑦 2 ) = 9

Q9:

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

  • A ( 𝑥 + 6 ) + ( 𝑦 2 ) = 1 8
  • B ( 𝑥 3 ) + ( 𝑦 + 1 ) = 8 1
  • C ( 𝑥 + 1 2 ) + ( 𝑦 1 ) = 1 6 2
  • D ( 𝑥 + 3 ) + ( 𝑦 1 ) = 8 1

Q10:

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

  • A 𝑥 + ( 𝑦 1 ) = 5 0
  • B ( 𝑥 4 ) + ( 𝑦 4 ) = 2 5
  • C ( 𝑥 8 ) + ( 𝑦 8 ) = 1 0
  • D ( 𝑥 + 4 ) + ( 𝑦 + 4 ) = 2 5

Q11:

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

  • A ( 𝑥 + 6 ) + ( 𝑦 + 6 ) = 1 0
  • B 𝑦 + ( 𝑥 + 7 ) = 5 0
  • C ( 𝑥 + 3 ) + ( 𝑦 + 3 ) = 2 5
  • D ( 𝑥 3 ) + ( 𝑦 3 ) = 2 5

Q12:

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

  • A ( 𝑥 8 ) + ( 𝑦 + 4 ) = 1 6
  • B ( 𝑥 + 4 ) + ( 𝑦 2 ) = 6 4
  • C ( 𝑥 4 ) + ( 𝑦 + 2 ) = 6 4
  • D ( 𝑥 + 4 ) + ( 𝑦 + 2 ) = 1 2 8

Q13:

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

  • A ( 𝑥 + 1 ) + ( 𝑦 + 1 ) = 2 5
  • B ( 𝑥 1 ) + ( 𝑦 1 ) = 2 5
  • C ( 𝑥 2 ) + ( 𝑦 2 ) = 1 0
  • D ( 𝑥 + 3 ) + ( 𝑦 + 2 ) = 5 0

Q14:

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

  • A ( 𝑥 + 8 ) + ( 𝑦 3 ) = 5 0
  • B ( 𝑥 + 3 ) + ( 𝑦 3 ) = 2 5
  • C ( 𝑥 3 ) + ( 𝑦 + 3 ) = 2 5
  • D ( 𝑥 + 6 ) + ( 𝑦 6 ) = 1 0

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 ( 𝑥 + 4 ) + ( 𝑦 + 4 ) = 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?

  • A2
  • Binfinite
  • C0
  • D1

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).

  • A 𝑥 + 𝑦 1 6 𝑥 2 0 𝑦 + 8 = 0
  • B 𝑥 + 𝑦 1 6 𝑥 2 0 𝑦 + 6 4 = 0
  • C 𝑥 + 𝑦 + 8 𝑥 + 1 0 𝑦 + 6 4 = 0
  • D 𝑥 + 𝑦 + 8 𝑥 + 1 6 𝑦 + 6 4 = 0

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