Worksheet: Equation of a Circle

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

Q1:

Write, in the form 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑥 + 𝑑 𝑦 + 𝑒 = 0 , the equation of the circle of radius 10 and center ( 4 , − 7 ) .

  • A 𝑥 + 𝑦 + 4 𝑥 − 7 𝑦 + 1 6 5 = 0
  • B 𝑥 + 𝑦 + 8 𝑥 − 1 4 𝑦 − 3 5 = 0
  • C 𝑥 + 𝑦 − 4 𝑥 + 7 𝑦 + 1 6 5 = 0
  • D 𝑥 + 𝑦 − 8 𝑥 + 1 4 𝑦 − 3 5 = 0

Q2:

Write the equation of the circle of centre ( 0 , 5 ) and diameter 10.

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

Q3:

Give the general form of the equation of the circle centre ( 8 , − 2 ) and diameter 10.

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

Q4:

Find the general form of the equation of circle 𝑀 , given that it touches the two coordinate axes at 𝐴 and 𝐵 and that 𝑀 𝑂 = 6 √ 2 .

  • A 𝑥 + 𝑦 − 6 𝑥 − 6 𝑦 + 3 6 = 0
  • B 𝑥 + 𝑦 + 1 2 𝑥 + 1 2 𝑦 = 0
  • C 𝑥 + 𝑦 − 1 2 𝑥 − 1 2 𝑦 = 0
  • D 𝑥 + 𝑦 − 1 2 𝑥 − 1 2 𝑦 + 3 6 = 0

Q5:

Write the equation of the circle of center ( 8 , 4 ) and radius 9.

  • A ( 𝑥 − 8 ) + ( 𝑦 − 4 ) = 9
  • B ( 𝑥 + 8 ) + ( 𝑦 + 4 ) = 8 1
  • C ( 𝑥 + 8 ) + ( 𝑦 + 4 ) = 9
  • D ( 𝑥 − 8 ) + ( 𝑦 − 4 ) = 8 1

Q6:

Determine the equation of a circle with radius = 1 7 c m , given that it touches the 𝑦 -axis at the point ( 0 , − 7 ) , and its centre lies in the third quadrant.

  • A ( 𝑥 − 1 7 ) + ( 𝑦 − 7 ) = 2 8 9 2 2
  • B 𝑥 + ( 𝑦 + 7 ) = 2 8 9 2 2
  • C ( 𝑥 + 7 ) + 𝑦 = 2 8 9 2 2
  • D ( 𝑥 + 1 7 ) + ( 𝑦 + 7 ) = 2 8 9 2 2

Q7:

What is the equation of the circle of radius 24 that lies in the third quadrant and is tangent to the two axes?

  • A 𝑥 + 𝑦 + 2 4 𝑥 + 2 4 𝑦 + 5 7 6 = 0 2 2
  • B 𝑥 + 𝑦 − 4 8 𝑥 − 4 8 𝑦 + 5 7 6 = 0 2 2
  • C 𝑥 + 𝑦 − 4 8 𝑥 + 4 8 𝑦 + 5 7 6 = 0 2 2
  • D 𝑥 + 𝑦 + 4 8 𝑥 + 4 8 𝑦 + 5 7 6 = 0 2 2

Q8:

Find the point of intersection between the line with equation 𝑦 = 1 2 5 𝑥 − 2 6 and the circle with center ( − 2 , 3 ) and radius 13.

  • A ( 3 , − 9 )
  • B ( − 2 , − 1 0 )
  • C ( 1 1 , 3 )
  • D ( 1 0 , − 2 )
  • E ( 2 5 , 3 4 )

Q9:

Let us consider a circle of radius 4 and center ( 2 , − 7 ) .

Write the equation of the circle.

  • A ( 𝑥 + 2 ) + ( 𝑦 − 7 ) = 4 2 2
  • B ( 𝑥 − 2 ) + ( 𝑦 + 7 ) = 4 2 2
  • C ( 𝑥 + 2 ) + ( 𝑦 − 7 ) = 1 6 2 2
  • D ( 𝑥 − 2 ) + ( 𝑦 + 7 ) = 1 6 2 2
  • E ( 𝑥 − 2 ) + ( 𝑦 − 7 ) = 1 6 2 2

The circle is dilated by a factor of 2. The center of dilation is the center of the circle. Write the equation of the circle.

  • A ( 𝑥 − 2 ) + ( 𝑦 + 7 ) = 6 4 2 2
  • B ( 𝑥 + 2 ) + ( 𝑦 − 7 ) = 6 4 2 2
  • C ( 𝑥 − 2 ) + ( 𝑦 + 7 ) = 8 2 2
  • D ( 𝑥 − 2 ) + ( 𝑦 + 7 ) = 3 2 2 2
  • E ( 𝑥 + 2 ) + ( 𝑦 − 7 ) = 3 2 2 2

Q10:

Let us consider a circle of radius 6 and center ( − 2 , − 5 ) .

Write the equation of the circle.

  • A ( 𝑥 + 2 ) + ( 𝑦 + 5 ) = 6 2 2
  • B ( 𝑥 − 2 ) + ( 𝑦 − 5 ) = 3 6 2 2
  • C ( 𝑥 − 2 ) + ( 𝑦 − 5 ) = 6 2 2
  • D ( 𝑥 + 2 ) + ( 𝑦 + 5 ) = 3 6 2 2

The circle is dilated by a factor of 1 3 . The center of dilation is the center of the circle. Write the equation of the circle.

  • A ( 𝑥 + 2 ) + ( 𝑦 + 5 ) = 4 2 2
  • B ( 𝑥 − 2 ) + ( 𝑦 − 5 ) = 4 2 2
  • C ( 𝑥 + 2 ) + ( 𝑦 + 5 ) = 2 2 2
  • D ( 𝑥 + 2 ) + ( 𝑦 + 5 ) = 1 2 2 2
  • E ( 𝑥 − 2 ) + ( 𝑦 − 5 ) = 1 2 2 2

Q11:

A circle is tangent to the 𝑥 -axis at ( 8 , 0 ) and cuts a chord of length 2 √ 3 7 7 on the negative 𝑦 -axis. What is the equation of the circle?

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

Q12:

A circle of radius 15 length units has its centre 𝑀 at the point ( − 6 , − 9 ) . Given that the circle intersects the 𝑥 -axis at points 𝐴 and 𝐵 , determine the area of △ 𝑀 𝐴 𝐵 .

Q13:

In the figure below, find the equation of the circle.

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

Q14:

Given 𝐴 ( 1 0 , 9 ) and 𝐵 ( 1 0 , − 1 ) , find the equation of the circle with diameter 𝐴 𝐵 .

  • A 𝑥 + 𝑦 − 2 0 𝑥 + 2 𝑦 + 7 6 = 0
  • B 𝑥 + 𝑦 − 2 0 𝑥 − 1 8 𝑦 + 1 5 6 = 0
  • C 𝑥 + 𝑦 − 2 0 𝑥 − 8 𝑦 + 1 6 = 0
  • D 𝑥 + 𝑦 − 2 0 𝑥 − 8 𝑦 + 9 1 = 0

Q15:

Write, in the form 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑥 + 𝑑 𝑦 + 𝑒 = 0 , the equation of the circle of radius 10 and center ( − 7 , − 8 ) .

  • A 𝑥 + 𝑦 − 7 𝑥 − 8 𝑦 + 2 1 3 = 0
  • B 𝑥 + 𝑦 − 1 4 𝑥 − 1 6 𝑦 + 1 3 = 0
  • C 𝑥 + 𝑦 + 7 𝑥 + 8 𝑦 + 2 1 3 = 0
  • D 𝑥 + 𝑦 + 1 4 𝑥 + 1 6 𝑦 + 1 3 = 0

Q16:

Write, in the form 𝑎 𝑥 + 𝑏 𝑦 + 𝑐 𝑥 + 𝑑 𝑦 + 𝑒 = 0 , the equation of the circle of radius 9 and center ( 8 , − 6 ) .

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

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