Worksheet: Application of Equations of Circles

In this worksheet, we will practice extracting the radius and center of a circle from its equation.

Q1:

By completing the square, find the centre and radius of the circle 𝑥 4 𝑥 + 𝑦 4 𝑦 8 = 0 2 2 .

  • A centre: ( 2 , 2 ) , radius: 4
  • B centre: ( 2 , 2 ) , radius: 16
  • C centre: ( 2 , 2 ) , radius: 4
  • D centre: ( 2 , 2 ) , radius: 4
  • Ecentre: ( 2 , 2 ) , radius: 16

Q2:

The equation ( 𝑥 3 ) + ( 𝑦 + 2 ) = 1 0 0 describes a circle. Find its center and radius.

  • A ( 3 , 2 ) , 1 0
  • B ( 3 , 2 ) , 1 0 0
  • C ( 3 , 2 ) , 1 0 0
  • D ( 3 , 2 ) , 1 0

Q3:

A circle is the translation of circle ( 𝑥 3 ) + ( 𝑦 1 ) = 1 0 0 by ( 9 , 4 ) . What is its equation?

  • A ( 𝑥 7 ) + ( 𝑦 + 8 ) = 1 0 0
  • B ( 𝑥 1 2 ) + ( 𝑦 + 3 ) = 1 0 0
  • C ( 𝑥 3 ) + ( 𝑦 1 ) = 1 0 0
  • D ( 𝑥 + 6 ) + ( 𝑦 5 ) = 1 0 0

Q4:

Given that the equation of the given circle is ( 𝑥 4 ) + ( 𝑦 9 ) = 2 2 5 , find the length of 𝐴 𝐵 .

Q5:

What are the relative positions of circles ( 𝑥 5 ) + ( 𝑦 + 1 4 ) = 9 2 2 and ( 𝑥 1 3 ) + ( 𝑦 + 1 4 ) = 3 6 2 2 ?

  • AThe two circles touch externally.
  • BThe two circles are disjoint.
  • CThe two circles touch internally.
  • DThe two circles intersect.

Q6:

In what interval must lie for 𝑥 + 𝑦 + 2 𝑥 4 𝑦 + 1 4 = 0 to be the equation of a circle?

  • A ( 2 , )
  • B ( , 9 ]
  • C [ 4 , 2 ]
  • D ( 9 , )

Q7:

Given 𝐴 ( 2 , 4 ) and 𝐵 ( 1 0 , 0 ) , determine the equation of the circle for which 𝐴 𝐵 is a diameter.

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

Q8:

Given that ( 𝑛 + 1 0 ) 𝑥 + 𝑦 1 6 𝑦 + ( 𝑚 + 2 3 ) 𝑥 𝑦 + ( 𝑚 𝑛 ) 𝑥 + 1 0 9 = 0 is a circle, what is its radius?

Q9:

Determine the equation of the circle that passes through ( 1 , 5 ) and ( 5 , 5 ) and has its centre on the 𝑥 -axis.

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

Q10:

Find the equation of the circle with center ( 2 , 7 ) which touches the 𝑥 -axis.

  • A ( 𝑥 2 ) + ( 𝑦 7 ) = 4
  • B ( 𝑥 + 2 ) + ( 𝑦 + 7 ) = 4 9
  • C ( 𝑥 + 2 ) + ( 𝑦 + 7 ) = 4
  • D ( 𝑥 2 ) + ( 𝑦 7 ) = 4 9

Q11:

The equation 7 𝑥 + 𝑎 𝑦 + 𝑏 𝑥 𝑦 9 = 0 represents a circle, what are 𝑎 and 𝑏 ?

  • A 𝑎 = 7 , 𝑏 = 1
  • B 𝑎 = 1 , 𝑏 = 0
  • C 𝑎 = 7 , 𝑏 = 7
  • D 𝑎 = 7 , 𝑏 = 0

Q12:

Find the equations of all possible circles that pass through the two points 𝐴 ( 7 , 0 ) and 𝐵 ( 1 , 0 ) and have a radius of 5.

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

Q13:

Does the equation 𝑥 + 𝑦 4 𝑥 8 𝑦 6 1 = 0 represent a circle?

  • Ayes
  • Bno

Q14:

Does the equation 𝑥 + 𝑦 1 6 𝑥 1 0 𝑦 + 1 8 9 = 0 represent a circle?

  • Ano
  • Byes

Q15:

Find all the real values of for which 𝑥 + 𝑦 + 4 𝑥 + 2 𝑦 + 7 3 = 0 is the equation of a circle.

  • A ( 8 , 3 )
  • B ( , )
  • C [ 8 , 1 ]
  • D ( 1 , 8 )

Q16:

The line 𝑦 = 𝑚 𝑥 is tangent to the circle ( 𝑥 5 ) + ( 𝑦 3 ) = 2 5 . What is 𝑚 ?

  • A5
  • B4
  • C 5 3
  • D 8 1 5

Q17:

The circle in the figure has equation 𝑥 + 𝑦 1 6 𝑥 + 2 𝑦 + 4 9 = 0 , and the line 𝐿 is 𝑥 + 𝑦 + 1 = 0 . Find the length of 𝐴 𝐵 to the nearest hundredth.

Q18:

Consider a circle centered at ( 1 , 1 ) passing through the origin.

Write the equation of the circle.

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

The circle is dilated by a scale factor of two such that the circle still passes through the origin. Write the equation of the circle.

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

What is the radius of the new circle?

  • A 8
  • B4
  • C 2 2
  • D8
  • E2

Q19:

Are the two circles expressed by the equations 𝑥 + 𝑦 + 1 4 𝑥 1 6 𝑦 + 6 4 = 0 and 𝑥 + 𝑦 + 1 4 𝑥 + 4 𝑦 + 3 7 = 0 congruent?

  • Ano
  • Byes

Q20:

Find the centre and radius of the circle 𝑥 + 4 𝑥 + 𝑦 + 1 2 𝑦 + 7 3 2 = 0 2 2 .

  • A centre: ( 2 , 6 ) , radius: 7 2
  • B centre: ( 2 , 6 ) , radius: 7 2
  • C centre: ( 2 , 6 ) , radius: 7 2
  • D centre: ( 2 , 6 ) , radius: 7 2
  • Ecentre: ( 6 , 2 ) , radius: 7 2

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