Worksheet: Equation of a Circle Passing through Three Points

In this worksheet, we will practice finding the equation of a circle passing through three given points.

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 2 2
  • B ( 𝑥 + 2 ) + ( 𝑦 + 3 ) = 2 2 2
  • C ( 𝑥 + 6 ) + ( 𝑦 + 6 ) = 2 2 2
  • D ( 𝑥 + 3 ) + ( 𝑦 + 3 ) = 1 2 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 , 1 0 ) .

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

Q3:

Find the general equation of the circle through the origin that also passes through ( 1 2 , 0 ) and ( 0 , 1 6 ) .

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

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

Q5:

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

  • A ( 2 , 0 )
  • B ( 1 . 5 , 0 . 5 )
  • C ( 0 , 1 . 5 )
  • D ( 1 , 0 . 5 )

Q6:

The points 𝐴 ( 1 , 1 ) , 𝐵 ( 1 , 5 ) , 𝐶 ( 1 7 , 1 1 ) , and 𝐷 ( 1 9 , 5 ) form a rectangle. What is the equation of the circle that contains all four points?

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

Q7:

The coordinates for three of a group of aerialists in a circular formation are 𝐺 ( 2 3 , 9 ) , 𝐻 ( 1 2 , 2 ) , and 𝐽 ( 1 2 , 2 0 ) . 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 + 2 ) + ( 𝑦 + 2 ) = 9 2 2
  • B ( 𝑥 2 ) + ( 𝑦 5 ) = 1 8 2 2
  • C ( 𝑥 4 ) + ( 𝑦 4 ) = 6 2 2
  • D ( 𝑥 2 ) + ( 𝑦 2 ) = 9 2 2

Q9:

Find the equation of the circle that passes through the points 𝐴 ( 6 , 1 ) , 𝐵 ( 3 , 1 0 ) , and 𝐶 ( 1 2 , 1 ) .

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

Q10:

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

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

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

Q12:

Find the equation of the circle that passes through the points 𝐴 ( 1 2 , 2 ) , 𝐵 ( 4 , 6 ) , and 𝐶 ( 4 , 2 ) .

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

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

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

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

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