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Worksheet: Equation of a Circle

Q1:

The equation ( π‘₯ βˆ’ 3 ) + ( 𝑦 + 2 ) = 1 0 0 2 2 describes a circle. Find its center and radius.

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

Q2:

By completing the square, find the center and radius of the circle π‘₯ βˆ’ 4 π‘₯ + 𝑦 βˆ’ 4 𝑦 βˆ’ 8 = 0 2 2 .

  • A center: ( βˆ’ 2 , βˆ’ 2 ) , radius: 4
  • B center: ( βˆ’ 2 , βˆ’ 2 ) , radius: 16
  • C center: ( 2 , βˆ’ 2 ) , radius: 4
  • D center: ( 2 , 2 ) , radius: 4
  • Ecenter: ( 2 , 2 ) , radius: 16

Q3:

A circle is the translation of circle by ( π‘₯ βˆ’ 3 ) + ( 𝑦 βˆ’ 1 ) = 1 0 0 2 2 by ( βˆ’ 9 , 4 ) . What is its equation?

  • A ( π‘₯ βˆ’ 7 ) + ( 𝑦 + 8 ) = 1 0 0 2 2
  • B ( π‘₯ βˆ’ 1 2 ) + ( 𝑦 + 3 ) = 1 0 0 2 2
  • C ( π‘₯ βˆ’ 3 ) + ( 𝑦 βˆ’ 1 ) = 1 0 0 2 2
  • D ( π‘₯ + 6 ) + ( 𝑦 βˆ’ 5 ) = 1 0 0 2 2

Q4:

A circle is the translation of circle by ( π‘₯ + 1 ) + ( 𝑦 + 1 ) = 4 9 2 2 by ( 4 , βˆ’ 5 ) . What is its equation?

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

Q5:

A circle is the translation of circle by ( π‘₯ + 3 ) + ( 𝑦 + 4 ) = 1 0 0 2 2 by ( βˆ’ 6 , βˆ’ 1 ) . What is its equation?

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

Q6:

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

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

Q7:

The circle in the figure has equation π‘₯ + 𝑦 βˆ’ 1 6 π‘₯ + 2 𝑦 + 4 9 = 0   , and the line 𝐿 is π‘₯ + 𝑦 + 1 = 0 . Find the length of 𝐴 𝐡 to the nearest hundredth.

Q8:

Given that ( 𝑛 + 1 0 ) π‘₯ + 𝑦 βˆ’ 1 6 𝑦 + ( π‘š + 2 3 ) π‘₯ 𝑦 + ( π‘š βˆ’ 𝑛 ) π‘₯ + 1 0 9 = 0 2 2 is a circle, what is its radius?

Q9:

Determine the equation of the circle that passes through ( 1 , βˆ’ 5 ) and ( 5 , βˆ’ 5 ) and has its center on the π‘₯ -axis.

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

Q10:

Find the equation of the circle whose center ( 2 , 7 ) , touching the π‘₯ -axis.

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

Q11:

What are the relative positions of circles ( π‘₯ βˆ’ 5 ) + ( 𝑦 + 1 4 ) = 9 2 2 and ( π‘₯ βˆ’ 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.

Q12:

The equation 7 π‘₯ + π‘Ž 𝑦 + 𝑏 π‘₯ 𝑦 βˆ’ 9 = 0 2 2 represents a circle, what are π‘Ž and 𝑏 ?

  • A π‘Ž = 7 , 𝑏 = 1
  • B π‘Ž = 1 , 𝑏 = 0
  • C π‘Ž = 7 , 𝑏 = 7
  • D π‘Ž = 7 , 𝑏 = 0

Q13:

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

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

Q14:

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

Write the equation of the circle.

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

What is the radius of the new circle?

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

Q15:

Does the equation π‘₯ + 𝑦 βˆ’ 4 π‘₯ βˆ’ 8 𝑦 βˆ’ 6 1 = 0 2 2 represent a circle?

  • Ayes
  • Bno

Q16:

Does the equation π‘₯ + 𝑦 βˆ’ 1 6 π‘₯ βˆ’ 1 0 𝑦 + 1 8 9 = 0 2 2 represent a circle?

  • Ano
  • Byes

Q17:

In what interval must lie for to be the equation of a circle?

  • A
  • B
  • C
  • D

Q18:

Find all the real values of for which is the equation of a circle.

  • A
  • B
  • C
  • D

Q19:

The line 𝑦 = π‘š π‘₯ is tangent to the circle ( π‘₯ βˆ’ 5 ) + ( 𝑦 βˆ’ 3 ) = 2 5 2 2 . What is π‘š ?

  • A5
  • B4
  • C βˆ’ 5 3
  • D βˆ’ 8 1 5

Q20:

Given that the equation of the given circle is ( π‘₯ βˆ’ 4 ) + ( 𝑦 βˆ’ 9 ) = 2 2 5   , find the length of 𝐴 𝐡 .