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Worksheet: Writing the Equation of a Circle Using Its Center and a Given Point

Q1:

Determine the equation of a circle that passes through the point 𝐴 ( 0 , 8 ) if its center is 𝑀 ( βˆ’ 2 , βˆ’ 6 ) .

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

Q2:

Determine the equation of a circle that passes through the point 𝐴 ( 5 , 1 0 ) if its center is 𝑀 ( 6 , 9 ) .

  • A ( π‘₯ βˆ’ 6 ) + ( 𝑦 βˆ’ 9 ) = √ 2
  • B ( π‘₯ + 6 ) + ( 𝑦 + 9 ) = 2 2 2
  • C ( π‘₯ + 6 ) + ( 𝑦 + 9 ) = √ 2
  • D ( π‘₯ βˆ’ 6 ) + ( 𝑦 βˆ’ 9 ) = 2 2 2

Q3:

Determine the equation of a circle that passes through the point 𝐴 ( 5 , βˆ’ 1 0 ) if its center is 𝑀 ( 2 , βˆ’ 4 ) .

  • A ( π‘₯ βˆ’ 2 ) + ( 𝑦 + 4 ) = 3 √ 5
  • B ( π‘₯ + 2 ) + ( 𝑦 βˆ’ 4 ) = 4 5 2 2
  • C ( π‘₯ + 2 ) + ( 𝑦 βˆ’ 4 ) = 3 √ 5
  • D ( π‘₯ βˆ’ 2 ) + ( 𝑦 + 4 ) = 4 5 2 2

Q4:

Determine the equation of a circle that passes through the point 𝐴 ( 1 , 3 ) if its center is 𝑀 ( 1 0 , βˆ’ 3 ) .

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

Q5:

Determine the equation of a circle that passes through the point 𝐴 ( 8 , βˆ’ 2 ) if its center is 𝑀 ( 5 , βˆ’ 8 ) .

  • A ( π‘₯ βˆ’ 5 ) + ( 𝑦 + 8 ) = 3 √ 5
  • B ( π‘₯ + 5 ) + ( 𝑦 βˆ’ 8 ) = 4 5 2 2
  • C ( π‘₯ + 5 ) + ( 𝑦 βˆ’ 8 ) = 3 √ 5
  • D ( π‘₯ βˆ’ 5 ) + ( 𝑦 + 8 ) = 4 5 2 2

Q6:

Determine the equation of a circle that passes through the point 𝐴 ( βˆ’ 7 , 2 ) if its center is 𝑀 ( βˆ’ 3 , 2 ) .

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

Q7:

Determine the equation of a circle that passes through the point 𝐴 ( 9 , βˆ’ 5 ) if its center is 𝑀 ( βˆ’ 6 , 1 0 ) .

  • A ( π‘₯ + 6 ) + ( 𝑦 βˆ’ 1 0 ) = 1 5 √ 2
  • B ( π‘₯ βˆ’ 6 ) + ( 𝑦 + 1 0 ) = 4 5 0 2 2
  • C ( π‘₯ βˆ’ 6 ) + ( 𝑦 + 1 0 ) = 1 5 √ 2
  • D ( π‘₯ + 6 ) + ( 𝑦 βˆ’ 1 0 ) = 4 5 0 2 2

Q8:

Determine the equation of a circle that passes through the point 𝐴 ( 4 , βˆ’ 1 0 ) if its center is 𝑀 ( βˆ’ 3 , βˆ’ 1 0 ) .

  • A ( π‘₯ + 3 ) + ( 𝑦 + 1 0 ) = 7
  • B ( π‘₯ βˆ’ 3 ) + ( 𝑦 βˆ’ 1 0 ) = 4 9 2 2
  • C ( π‘₯ βˆ’ 3 ) + ( 𝑦 βˆ’ 1 0 ) = 7
  • D ( π‘₯ + 3 ) + ( 𝑦 + 1 0 ) = 4 9 2 2

Q9:

Determine the equation of a circle that passes through the point 𝐴 ( 3 , βˆ’ 4 ) if its center is 𝑀 ( 0 , βˆ’ 5 ) .

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

Q10:

Determine the equation of a circle that passes through the point 𝐴 ( 7 , βˆ’ 5 ) if its center is 𝑀 ( 7 , 2 ) .

  • A ( π‘₯ βˆ’ 7 ) + ( 𝑦 βˆ’ 2 ) = 7
  • B ( π‘₯ + 7 ) + ( 𝑦 + 2 ) = 4 9 2 2
  • C ( π‘₯ + 7 ) + ( 𝑦 + 2 ) = 7
  • D ( π‘₯ βˆ’ 7 ) + ( 𝑦 βˆ’ 2 ) = 4 9 2 2

Q11:

A circle has center ( 2 , 2 ) and goes through the point ( 6 , 3 ) . Find the equation of the circle.

  • A ( π‘₯ + 2 ) βˆ’ ( 𝑦 + 2 ) = 1 7 2 2
  • B ( π‘₯ βˆ’ 2 ) + ( 𝑦 βˆ’ 2 ) = √ 1 7 2 2
  • C ( π‘₯ + 2 ) βˆ’ ( 𝑦 + 2 ) = √ 1 7 2 2
  • D ( π‘₯ βˆ’ 2 ) + ( 𝑦 βˆ’ 2 ) = 1 7 2 2
  • E ( π‘₯ βˆ’ 2 ) + ( 𝑦 + 2 ) = 1 7 2 2

Q12:

A circle has center ( 4 , βˆ’ 2 ) and goes through the point ( βˆ’ 2 , βˆ’ 3 ) . Find the equation of the circle.

  • A ( π‘₯ βˆ’ 4 ) + ( 𝑦 + 2 ) = √ 3 7 2 2
  • B ( π‘₯ + 2 ) + ( 𝑦 βˆ’ 2 ) = 3 7 2 2
  • C ( π‘₯ + 2 ) + ( 𝑦 βˆ’ 4 ) = √ 3 7 2 2
  • D ( π‘₯ βˆ’ 4 ) + ( 𝑦 + 2 ) = 3 7 2 2
  • E ( π‘₯ + 2 ) + ( 𝑦 βˆ’ 4 ) = 3 7 2 2

Q13:

A circle has center ο€Ό 2 3 , βˆ’ 2 5  and goes through the point ( βˆ’ 3 , 5 ) . Find the equation of the circle.

  • A ο€Ό π‘₯ + 2 3  + ο€Ό 𝑦 βˆ’ 2 5  = √ 9 5 8 6 1 5 2 2
  • B ( π‘₯ + 3 ) + ( 𝑦 βˆ’ 5 ) = 9 5 8 6 2 2 5 2 2
  • C ( π‘₯ βˆ’ 3 ) + ( 𝑦 + 5 ) = √ 9 5 8 6 1 5 2 2
  • D ο€Ό π‘₯ βˆ’ 2 3  + ο€Ό 𝑦 + 2 5  = 9 5 8 6 2 2 5 2 2
  • E ( π‘₯ + 2 ) βˆ’ ( 𝑦 βˆ’ 5 ) = 9 5 8 6 2 2 5 2 2

Q14:

A circle centered at the origin goes through the point (1, 1).

Work out the equation of the circle.

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

Determine the value of 𝑦 when π‘₯ = 1 2 .

  • A 𝑦 = √ 7 2
  • B 𝑦 = ο„ž 7 2
  • C 𝑦 = ο„ž 3 2
  • D 𝑦 = √ 3 2
  • E 𝑦 = 3 2

Is the point ο€Ώ 1 2 , √ 7 2  on the circle?

  • Ano
  • Byes

Q15:

Determine the equation of a circle whose centre is at the point 𝑀 ( 4 , βˆ’ 3 ) , given that the circle touches the straight line π‘₯ = 1 0 .

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

Q16:

The given figure shows a circle with center 𝑂 ( π‘₯ , 𝑦 ) 0 0 and a point 𝐴 ( π‘₯ , 𝑦 ) lying on the circumference of the circle.

Find the length of 𝑂 𝐡 in terms of π‘₯ and π‘₯ 0 .

  • A π‘₯ + π‘₯ 0
  • B √ π‘₯ βˆ’ π‘₯ 0
  • C 𝑦 βˆ’ 𝑦 0
  • D π‘₯ βˆ’ π‘₯ 0
  • E √ 𝑦 βˆ’ 𝑦 0

Find the length of 𝐴 𝐡 in terms of 𝑦 and 𝑦 0 .

  • A 𝑦 βˆ’ 𝑦 0
  • B π‘₯ + π‘₯ 0
  • C √ π‘₯ βˆ’ π‘₯ 0
  • D √ 𝑦 βˆ’ 𝑦 0
  • E π‘₯ βˆ’ π‘₯ 0

Using the Pythagorean Theorem, express π‘Ÿ 2 in terms of the lengths of 𝑂 𝐡 and 𝐴 𝐡 .

  • A ( π‘₯ βˆ’ 𝑦 ) + ( π‘₯ βˆ’ 𝑦 ) = π‘Ÿ 2 0 0 2 2
  • B ( π‘₯ + π‘₯ ) + ( 𝑦 + 𝑦 ) = π‘Ÿ 0 2 0 2 2
  • C ( π‘₯ βˆ’ π‘₯ ) + ( 𝑦 βˆ’ 𝑦 ) = π‘Ÿ 0 2 0 2 2
  • D ( π‘₯ βˆ’ π‘₯ ) βˆ’ ( 𝑦 βˆ’ 𝑦 ) = π‘Ÿ 0 2 0 2 2
  • E ( π‘₯ + π‘₯ ) + ( 𝑦 + 𝑦 ) = π‘Ÿ 0 2 0 2 2