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Lesson: Equation of a Circle with a Center and a Point

Sample Question Videos

Worksheet • 16 Questions • 4 Videos

Q1:

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

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

Q2:

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

  • A ( π‘₯ + 2 ) + ( 𝑦 + 6 ) = 2 0 0 2 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 ) = 1 0 √ 2

Q3:

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

Q4:

What is the equation of the circle of centre ( 3 , 4 ) and passing through ( 7 , 7 ) ?

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

Q5:

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

Q6:

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

Work out the equation of the circle.

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

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

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

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

  • ANo
  • BYes

Q7:

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 ) = 3 6 2 2
  • 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 ) = 6

Q8:

The given figure shows a circle with center 𝑂 ( π‘₯ , 𝑦 )   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‘₯  .

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

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

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

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

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

Q9:

Find the radius of the circle that passes through point ( βˆ’ 3 , βˆ’ 2 ) and has centre ( βˆ’ 5 , 8 ) .

  • A 2 √ 2 6
  • B21
  • C 2 √ 3
  • D √ 1 0 2
  • E √ 1 4

Q10:

Does the coordinate ( 3 , βˆ’ 1 ) lie on the circle centred at the point ( 2 , βˆ’ 2 ) passing through the point ( 1 , βˆ’ 1 ) ?

  • A no
  • B yes
  • C We need more information in order to work it out.

Q11:

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

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

Q12:

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

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

Q13:

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

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

Q14:

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

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

Q15:

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

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

Q16:

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

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