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

In this worksheet, we will practice finding the equation of a circle using its center and a given point.

Q1:

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

  • A ( 𝑥 + 2 ) ( 𝑦 + 2 ) = 1 7 2 2
  • B ( 𝑥 2 ) + ( 𝑦 2 ) = 1 7 2 2
  • C ( 𝑥 + 2 ) ( 𝑦 + 2 ) = 1 7 2 2
  • D ( 𝑥 2 ) + ( 𝑦 2 ) = 1 7 2 2
  • E ( 𝑥 2 ) + ( 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 ( 2 , 6 ) .

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

Q3:

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

  • A ( 𝑥 4 ) + ( 𝑦 + 2 ) = 3 7 2 2
  • B ( 𝑥 + 2 ) + ( 𝑦 2 ) = 3 7 2 2
  • C ( 𝑥 + 2 ) + ( 𝑦 4 ) = 3 7 2 2
  • D ( 𝑥 4 ) + ( 𝑦 + 2 ) = 3 7 2 2
  • E ( 𝑥 + 2 ) + ( 𝑦 4 ) = 3 7 2 2

Q4:

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

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

Q5:

A circle has centre 2 3 , 2 5 and goes through the point ( 3 , 5 ) . Find the equation of the circle.

  • A 𝑥 + 2 3 + 𝑦 2 5 = 9 5 8 6 1 5 2 2
  • B ( 𝑥 + 3 ) + ( 𝑦 5 ) = 9 5 8 6 2 2 5 2 2
  • C ( 𝑥 3 ) + ( 𝑦 + 5 ) = 9 5 8 6 1 5 2 2
  • D 𝑥 2 3 + 𝑦 + 2 5 = 9 5 8 6 2 2 5 2 2
  • E ( 𝑥 + 2 ) ( 𝑦 5 ) = 9 5 8 6 2 2 5 2 2

Q6:

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

Work out the equation of the circle.

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

Determine the positive value of 𝑦 when 𝑥 = 1 2 .

  • A 𝑦 = 7 2
  • B 𝑦 = 7 2
  • C 𝑦 = 3 2
  • D 𝑦 = 3 2
  • E 𝑦 = 3 2

Is the point 1 2 , 7 2 on the circle?

  • ANo
  • BYes

Q7:

Determine the equation of a circle whose centre is at the point 𝑀 ( 4 , 3 ) , given that the circle touches the straight line 𝑥 = 1 0 .

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

Q8:

The given figure shows a circle with center 𝑂 ( 𝑥 , 𝑦 ) and a point 𝐴 ( 𝑥 , 𝑦 ) lying on the circumference of the circle.

Find the length of 𝑂 𝐵 in terms of 𝑥 and 𝑥 .

  • A 𝑥 + 𝑥
  • B 𝑥 𝑥
  • C 𝑦 𝑦
  • D 𝑥 𝑥
  • E 𝑦 𝑦

Find the length of 𝐴 𝐵 in terms of 𝑦 and 𝑦 .

  • A 𝑦 𝑦
  • B 𝑥 + 𝑥
  • C 𝑥 𝑥
  • D 𝑦 𝑦
  • E 𝑥 𝑥

Using the Pythagorean Theorem, express 𝑟 in terms of the lengths of 𝑂 𝐵 and 𝐴 𝐵 .

  • A ( 𝑥 𝑦 ) + ( 𝑥 𝑦 ) = 𝑟
  • B ( 𝑥 + 𝑥 ) + ( 𝑦 + 𝑦 ) = 𝑟
  • C ( 𝑥 𝑥 ) + ( 𝑦 𝑦 ) = 𝑟
  • D ( 𝑥 𝑥 ) ( 𝑦 𝑦 ) = 𝑟
  • E ( 𝑥 + 𝑥 ) + ( 𝑦 + 𝑦 ) = 𝑟

Q9:

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

  • A 1 4
  • B 2 3
  • C 1 0 2
  • D 2 2 6
  • E21

Q10:

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

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

Q11:

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

  • A ( 𝑥 6 ) + ( 𝑦 9 ) = 2
  • B ( 𝑥 + 6 ) + ( 𝑦 + 9 ) = 2 2 2
  • C ( 𝑥 + 6 ) + ( 𝑦 + 9 ) = 2
  • D ( 𝑥 6 ) + ( 𝑦 9 ) = 2 2 2

Q12:

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

  • A ( 𝑥 2 ) + ( 𝑦 + 4 ) = 3 5
  • B ( 𝑥 + 2 ) + ( 𝑦 4 ) = 4 5 2 2
  • C ( 𝑥 + 2 ) + ( 𝑦 4 ) = 3 5
  • D ( 𝑥 2 ) + ( 𝑦 + 4 ) = 4 5 2 2

Q13:

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

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

Q14:

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

  • A ( 𝑥 5 ) + ( 𝑦 + 8 ) = 3 5
  • B ( 𝑥 + 5 ) + ( 𝑦 8 ) = 4 5 2 2
  • C ( 𝑥 + 5 ) + ( 𝑦 8 ) = 3 5
  • D ( 𝑥 5 ) + ( 𝑦 + 8 ) = 4 5 2 2

Q15:

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

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

Q16:

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

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

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