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 a given point or the radius and vice versa.

Q1:

Write, in the form 𝑎𝑥+𝑏𝑦+𝑐𝑥+𝑑𝑦+𝑒=0, the equation of the circle of radius 10 and center (4,−7).

  • A 𝑥 + 𝑦 + 8 𝑥 − 1 4 𝑦 − 3 5 = 0
  • B 𝑥 + 𝑦 − 4 𝑥 + 7 𝑦 + 1 6 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 center (0,5) and diameter 10.

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

Q3:

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

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

Q4:

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

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

Q5:

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

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

Q6:

Determine the equation of a circle with radius =17cm, given that it touches the 𝑦-axis at the point (0,−7), and its center lies in the third quadrant.

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

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

Q8:

Find the point of intersection between the line with equation 𝑦=125𝑥−26 and the circle with center (−2,3) and radius 13.

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

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

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

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

The circle is dilated by a factor of 13. The center of dilation is the center of the circle. Write the equation of the circle.

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

Q11:

A circle is tangent to the 𝑥-axis at (8,0) and cuts a chord of length 2√377 on the negative𝑦-axis. What is the equation of the circle?

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

Q13:

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

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

Q14:

Given 𝐴(10,9) and 𝐵(10,−1), find the equation of the circle with diameter 𝐴𝐵.

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

Q15:

Write, in the form 𝑎𝑥+𝑏𝑦+𝑐𝑥+𝑑𝑦+𝑒=0, the equation of the circle of radius 10 and center (−7,−8).

  • A 𝑥 + 𝑦 − 1 4 𝑥 − 1 6 𝑦 + 1 3 = 0
  • B 𝑥 + 𝑦 + 7 𝑥 + 8 𝑦 + 2 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 𝑎𝑥+𝑏𝑦+𝑐𝑥+𝑑𝑦+𝑒=0, the equation of the circle of radius 9 and center (8,−6).

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

Q17:

Let us consider a circle of radius 5 and center (4,−8).

Write the equation of the circle.

  • A ( 𝑥 − 4 ) + ( 𝑦 + 8 ) = 5
  • B ( 𝑥 − 4 ) + ( 𝑦 + 8 ) = 2 5
  • C ( 𝑥 + 4 ) + ( 𝑦 − 8 ) = 5
  • D ( 𝑥 + 4 ) + ( 𝑦 − 8 ) = 2 5
  • E ( 𝑥 − 4 ) + ( 𝑦 + 8 ) = √ 5

The circle undergoes a dilation by a scale factor of three, centered at (4,−8), and then a translation six units to the left and three units up. Write the equation of the circle.

  • A ( 𝑥 − 1 0 ) + ( 𝑦 + 1 1 ) = 7 5
  • B ( 𝑥 + 2 ) + ( 𝑦 + 5 ) = 7 5
  • C ( 𝑥 + 2 ) + ( 𝑦 + 5 ) = 1 5
  • D ( 𝑥 + 2 ) + ( 𝑦 + 5 ) = 2 2 5
  • E ( 𝑥 − 1 0 ) + ( 𝑦 + 1 1 ) = 2 2 5

Q18:

Determine the equation of a circle with a diameter of 14 feet whose center was translated 15 feet left and 14 feet up from the origin.

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

Q19:

Find the equation of a circle which has the radius as the circle 𝑥+𝑦+18𝑥𝜃+18𝑦𝜃+17=0cossin, and two of whose diameters lie on lines 9𝑥+4𝑦+50=0 and r=⟨−4,−1⟩+𝑠⟨1,−1⟩.

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

Q20:

Determine the general form of the equation of the circle that passes through the two points 𝐴(−7,1) and 𝐵(0,6), given that the circle’s center lies on the straight line 6𝑥−𝑦=−43.

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

Q21:

Find the general form of the equation of the circle 𝑀 if the straight line 𝐿 of equation 2𝑥−3𝑦=0 passes through the center of the circle and the origin.

  • A 𝑥 + 𝑦 + 6 𝑥 + 4 𝑦 + 9 = 0
  • B 𝑥 + 𝑦 − 4 𝑥 − 6 𝑦 + 4 = 0
  • C 𝑥 + 𝑦 + 4 𝑥 + 6 𝑦 + 9 = 0
  • D 𝑥 + 𝑦 + 6 𝑥 + 4 𝑦 + 4 = 0

Q22:

A circle 𝑀 has circumference 26𝜋 and intersects the 𝑥-axis at points (−19,0) and (5,0). What are the possible equations for 𝑀?

  • A ( 𝑥 + 7 ) + ( 𝑦 + 5 ) = 1 3 , ( 𝑥 + 7 ) + ( 𝑦 − 5 ) = 1 3
  • B ( 𝑥 − 7 ) + ( 𝑦 − 5 ) = 1 6 9 , ( 𝑥 − 7 ) + ( 𝑦 + 5 ) = 1 6 9
  • C ( 𝑥 − 7 ) + ( 𝑦 − 5 ) = 1 3 , ( 𝑥 − 7 ) + ( 𝑦 + 5 ) = 1 3
  • D ( 𝑥 + 7 ) + ( 𝑦 + 5 ) = 1 6 9 , ( 𝑥 + 7 ) + ( 𝑦 − 5 ) = 1 6 9

Q23:

In the figure below, we are given that 𝐴(9,0) and 𝐵(0,18). Determine the equation of the circle at 𝑀.

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

Q24:

Find the center and radius of the circle 𝑥+3𝑥+𝑦+83𝑦−3,45536=0.

  • Acenter: 32,43, radius: 10
  • Bcenter: 32,−43, radius: 100
  • Ccenter: −43,32, radius: 100
  • Dcenter: −32,43, radius: 100
  • Ecenter: −32,−43, radius: 10

Q25:

By completing the square, find the center and radius of the circle 𝑥+6𝑥+𝑦−4𝑦+8=0.

  • Acenter: (2,−3), radius: 5
  • Bcenter: (−3,2), radius: 5
  • Ccenter: (3,−2), radius: 5
  • Dcenter: (2,−3), radius: √5
  • Ecenter: (−3,2), radius: √5

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