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Worksheet: Equation of a Circle Passing through Three Points

Q1:

Find the equation of the circle that passes through the points 𝐴 ( βˆ’ 4 , βˆ’ 3 ) , 𝐡 ( βˆ’ 3 , βˆ’ 4 ) , and 𝐢 ( βˆ’ 2 , βˆ’ 3 ) .

  • A ( π‘₯ βˆ’ 3 ) + ( 𝑦 βˆ’ 3 ) = 1 2 2
  • B ( π‘₯ + 2 ) + ( 𝑦 + 3 ) = 2 2 2
  • C ( π‘₯ + 6 ) + ( 𝑦 + 6 ) = 2 2 2
  • D ( π‘₯ + 3 ) + ( 𝑦 + 3 ) = 1 2 2

Q2:

Find the equation of the circle that passes through the points 𝐴 ( 2 , βˆ’ 1 ) , 𝐡 ( 5 , 2 ) , and 𝐢 ( 2 , 5 ) .

  • A ( π‘₯ + 2 ) + ( 𝑦 + 2 ) = 9 2 2
  • B ( π‘₯ βˆ’ 2 ) + ( 𝑦 βˆ’ 5 ) = 1 8 2 2
  • C ( π‘₯ βˆ’ 4 ) + ( 𝑦 βˆ’ 4 ) = 6 2 2
  • D ( π‘₯ βˆ’ 2 ) + ( 𝑦 βˆ’ 2 ) = 9 2 2

Q3:

Find the equation of the circle that passes through the points 𝐴 ( 6 , 1 ) , 𝐡 ( βˆ’ 3 , 1 0 ) , and 𝐢 ( βˆ’ 1 2 , 1 ) .

  • A ( π‘₯ βˆ’ 3 ) + ( 𝑦 + 1 ) = 8 1 2 2
  • B ( π‘₯ + 1 2 ) + ( 𝑦 βˆ’ 1 ) = 1 6 2 2 2
  • C ( π‘₯ + 6 ) + ( 𝑦 βˆ’ 2 ) = 1 8 2 2
  • D ( π‘₯ + 3 ) + ( 𝑦 βˆ’ 1 ) = 8 1 2 2

Q4:

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

  • A ( π‘₯ + 4 ) + ( 𝑦 + 4 ) = 2 5 2 2
  • B π‘₯ + ( 𝑦 βˆ’ 1 ) = 5 0 2 2
  • C ( π‘₯ βˆ’ 8 ) + ( 𝑦 βˆ’ 8 ) = 1 0 2 2
  • D ( π‘₯ βˆ’ 4 ) + ( 𝑦 βˆ’ 4 ) = 2 5 2 2

Q5:

Find the equation of the circle that passes through the points 𝐴 ( 1 , βˆ’ 6 ) , 𝐡 ( 0 , 1 ) , and 𝐢 ( βˆ’ 7 , 0 ) .

  • A ( π‘₯ βˆ’ 3 ) + ( 𝑦 βˆ’ 3 ) = 2 5 2 2
  • B 𝑦 + ( π‘₯ + 7 ) = 5 0 2 2
  • C ( π‘₯ + 6 ) + ( 𝑦 + 6 ) = 1 0 2 2
  • D ( π‘₯ + 3 ) + ( 𝑦 + 3 ) = 2 5 2 2

Q6:

Find the equation of the circle that passes through the points 𝐴 ( 1 2 , βˆ’ 2 ) , 𝐡 ( 4 , 6 ) , and 𝐢 ( βˆ’ 4 , βˆ’ 2 ) .

  • A ( π‘₯ + 4 ) + ( 𝑦 βˆ’ 2 ) = 6 4 2 2
  • B ( π‘₯ + 4 ) + ( 𝑦 + 2 ) = 1 2 8 2 2
  • C ( π‘₯ βˆ’ 8 ) + ( 𝑦 + 4 ) = 1 6 2 2
  • D ( π‘₯ βˆ’ 4 ) + ( 𝑦 + 2 ) = 6 4 2 2

Q7:

Find the equation of the circle that passes through the points 𝐴 ( 5 , 4 ) , 𝐡 ( βˆ’ 2 , 5 ) , and 𝐢 ( βˆ’ 3 , βˆ’ 2 ) .

  • A ( π‘₯ + 1 ) + ( 𝑦 + 1 ) = 2 5 2 2
  • B ( π‘₯ + 3 ) + ( 𝑦 + 2 ) = 5 0 2 2
  • C ( π‘₯ βˆ’ 2 ) + ( 𝑦 βˆ’ 2 ) = 1 0 2 2
  • D ( π‘₯ βˆ’ 1 ) + ( 𝑦 βˆ’ 1 ) = 2 5 2 2

Q8:

Find the equation of the circle that passes through the points 𝐴 ( 2 , 3 ) , 𝐡 ( βˆ’ 3 , 8 ) , and 𝐢 ( βˆ’ 8 , 3 ) .

  • A ( π‘₯ βˆ’ 3 ) + ( 𝑦 + 3 ) = 2 5 2 2
  • B ( π‘₯ + 8 ) + ( 𝑦 βˆ’ 3 ) = 5 0 2 2
  • C ( π‘₯ + 6 ) + ( 𝑦 βˆ’ 6 ) = 1 0 2 2
  • D ( π‘₯ + 3 ) + ( 𝑦 βˆ’ 3 ) = 2 5 2 2

Q9:

Find the equation of the circle that passes through the points 𝐴 ( βˆ’ 4 , βˆ’ 2 ) , 𝐡 ( βˆ’ 6 , βˆ’ 4 ) , and 𝐢 ( βˆ’ 4 , βˆ’ 6 ) .

  • A ( π‘₯ βˆ’ 4 ) + ( 𝑦 βˆ’ 4 ) = 4 2 2
  • B ( π‘₯ + 4 ) + ( 𝑦 + 6 ) = 8 2 2
  • C ( π‘₯ + 8 ) + ( 𝑦 + 8 ) = 4 2 2
  • D ( π‘₯ + 4 ) + ( 𝑦 + 4 ) = 4 2 2

Q10:

Find the equation of the circle that passes through the points 𝐴 ( βˆ’ 2 , 3 ) , 𝐡 ( βˆ’ 4 , 5 ) , and 𝐢 ( βˆ’ 6 , 3 ) .

  • A ( π‘₯ βˆ’ 4 ) + ( 𝑦 + 3 ) = 4 2 2
  • B ( π‘₯ + 6 ) + ( 𝑦 βˆ’ 3 ) = 8 2 2
  • C ( π‘₯ + 8 ) + ( 𝑦 βˆ’ 6 ) = 4 2 2
  • D ( π‘₯ + 4 ) + ( 𝑦 βˆ’ 3 ) = 4 2 2

Q11:

The points 𝐴 ( 1 , βˆ’ 1 ) , 𝐡 ( βˆ’ 1 , 5 ) , 𝐢 ( 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 ( π‘₯ + 9 ) + ( 𝑦 βˆ’ 5 ) = 3 6 0 2 2
  • B ( π‘₯ + 9 ) + ( 𝑦 + 5 ) = 4 0 0 2 2
  • C ( π‘₯ βˆ’ 9 ) + ( 𝑦 + 5 ) = 4 0 2 2
  • D ( π‘₯ βˆ’ 9 ) + ( 𝑦 βˆ’ 5 ) = 1 0 0 2 2

Q12:

Determine the general equation of the shown circle 𝑀 passing through the origin point and the two points 𝐴 ( 8 , 0 ) and 𝐡 ( 0 , βˆ’ 1 0 ) .

  • A π‘₯ + 𝑦 + 1 0 π‘₯ βˆ’ 8 𝑦 = 0 2 2
  • B π‘₯ + 𝑦 + 8 π‘₯ βˆ’ 1 0 𝑦 = 0 2 2
  • C π‘₯ + 𝑦 βˆ’ 1 6 π‘₯ + 2 0 𝑦 = 0 2 2
  • D π‘₯ + 𝑦 βˆ’ 8 π‘₯ + 1 0 𝑦 = 0 2 2

Q13:

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

  • A π‘₯ + 𝑦 βˆ’ 6 π‘₯ βˆ’ 8 𝑦 = 0 2 2
  • B π‘₯ + 𝑦 βˆ’ 2 4 π‘₯ βˆ’ 3 2 𝑦 + 3 0 0 = 0 2 2
  • C π‘₯ + 𝑦 + 1 2 π‘₯ + 1 6 𝑦 = 0 2 2
  • D π‘₯ + 𝑦 βˆ’ 1 2 π‘₯ βˆ’ 1 6 𝑦 = 0 2 2

Q14:

Find the general form of the equation of a circle that touches the π‘₯ -axis and passes through the two points ( βˆ’ 6 , βˆ’ 9 ) and ( 1 , βˆ’ 2 ) .

  • A π‘₯ + 𝑦 + 6 π‘₯ + 1 0 𝑦 + 2 5 = 0 2 2 , π‘₯ + 𝑦 βˆ’ 1 8 π‘₯ + 3 4 𝑦 + 2 8 9 = 0 2 2
  • B π‘₯ + 𝑦 + 6 π‘₯ + 1 0 𝑦 + 6 = 0 2 2 , π‘₯ + 𝑦 βˆ’ 1 8 π‘₯ + 3 4 𝑦 + 8 1 = 0 2 2
  • C π‘₯ + 𝑦 + 3 π‘₯ + 5 𝑦 + 9 = 0 2 2 , π‘₯ + 𝑦 βˆ’ 1 8 π‘₯ + 3 4 𝑦 + 8 1 = 0 2 2
  • D π‘₯ + 𝑦 + 6 π‘₯ + 1 0 𝑦 + 9 = 0 2 2 , π‘₯ + 𝑦 βˆ’ 1 8 π‘₯ + 3 4 𝑦 + 8 1 = 0 2 2

Q15:

Find the center of the circle through points 𝐴 ( 3 , 1 ) , 𝐡 ( 1 , 2 ) , and 𝐢 ( βˆ’ 1 , βˆ’ 2 ) .

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