Worksheet: Equation of a Circle

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 π‘Žπ‘₯+𝑏𝑦+𝑐π‘₯+𝑑𝑦+𝑒=0, the equation of the circle of radius 10 and center (4,βˆ’7).

  • Aπ‘₯+𝑦+8π‘₯βˆ’14π‘¦βˆ’35=0
  • Bπ‘₯+π‘¦βˆ’4π‘₯+7𝑦+165=0
  • Cπ‘₯+𝑦+4π‘₯βˆ’7𝑦+165=0
  • Dπ‘₯+π‘¦βˆ’8π‘₯+14π‘¦βˆ’35=0

Q2:

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

  • Aπ‘₯+(π‘¦βˆ’5)=5
  • Bπ‘₯+(𝑦+5)=5
  • Cπ‘₯+(π‘¦βˆ’5)=25
  • Dπ‘₯+(𝑦+5)=25

Q3:

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

  • Aπ‘₯+π‘¦βˆ’16π‘₯+4π‘¦βˆ’32=0
  • Bπ‘₯+𝑦+16π‘₯βˆ’4π‘¦βˆ’32=0
  • Cπ‘₯+π‘¦βˆ’16π‘₯+4𝑦+43=0
  • Dπ‘₯+𝑦+16π‘₯βˆ’4𝑦+43=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 𝐡 and that 𝑀𝑂=6√2.

  • Aπ‘₯+π‘¦βˆ’12π‘₯βˆ’12𝑦+36=0
  • Bπ‘₯+𝑦+12π‘₯+12𝑦=0
  • Cπ‘₯+π‘¦βˆ’12π‘₯βˆ’12𝑦=0
  • Dπ‘₯+π‘¦βˆ’6π‘₯βˆ’6𝑦+36=0

Q5:

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

  • A(π‘₯βˆ’8)+(π‘¦βˆ’4)=81
  • B(π‘₯βˆ’8)+(π‘¦βˆ’4)=9
  • C(π‘₯+8)+(𝑦+4)=9
  • D(π‘₯+8)+(𝑦+4)=81

Q6:

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

  • Aπ‘₯+(𝑦+7)=289
  • B(π‘₯+17)+(𝑦+7)=289
  • C(π‘₯βˆ’17)+(π‘¦βˆ’7)=289
  • D(π‘₯+7)+𝑦=289

Q7:

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

  • Aπ‘₯+𝑦+48π‘₯+48𝑦+576=0
  • Bπ‘₯+π‘¦βˆ’48π‘₯βˆ’48𝑦+576=0
  • Cπ‘₯+π‘¦βˆ’48π‘₯+48𝑦+576=0
  • Dπ‘₯+𝑦+24π‘₯+24𝑦+576=0

Q8:

Find the point of intersection between the line with equation 𝑦=125π‘₯βˆ’26 and the circle with center (βˆ’2,3) and radius 13.

  • A(11,3)
  • B(βˆ’2,βˆ’10)
  • C(25,34)
  • D(3,βˆ’9)
  • E(10,βˆ’2)

Q9:

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

Write the equation of the circle.

  • A(π‘₯+2)+(π‘¦βˆ’7)=4
  • B(π‘₯βˆ’2)+(π‘¦βˆ’7)=16
  • C(π‘₯βˆ’2)+(𝑦+7)=16
  • D(π‘₯+2)+(π‘¦βˆ’7)=16
  • E(π‘₯βˆ’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(π‘₯+2)+(π‘¦βˆ’7)=64
  • B(π‘₯+2)+(π‘¦βˆ’7)=32
  • C(π‘₯βˆ’2)+(𝑦+7)=64
  • D(π‘₯βˆ’2)+(𝑦+7)=32
  • E(π‘₯βˆ’2)+(𝑦+7)=8

Q10:

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

Write the equation of the circle.

  • A(π‘₯+2)+(𝑦+5)=6
  • B(π‘₯+2)+(𝑦+5)=36
  • C(π‘₯βˆ’2)+(π‘¦βˆ’5)=36
  • D(π‘₯βˆ’2)+(π‘¦βˆ’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(π‘₯+2)+(𝑦+5)=2
  • B(π‘₯βˆ’2)+(π‘¦βˆ’5)=4
  • C(π‘₯+2)+(𝑦+5)=12
  • D(π‘₯βˆ’2)+(π‘¦βˆ’5)=12
  • E(π‘₯+2)+(𝑦+5)=4

Q11:

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

  • Aπ‘₯+π‘¦βˆ’16π‘₯+42𝑦+64=0
  • Bπ‘₯+π‘¦βˆ’16π‘₯+42𝑦+128=0
  • Cπ‘₯+π‘¦βˆ’16π‘₯+42𝑦+441=0
  • Dπ‘₯+π‘¦βˆ’16π‘₯+42π‘¦βˆ’1,003=0

Q12:

A circle of radius 15 length units has its center 𝑀 at the point (βˆ’6,βˆ’9). Given that the circle intersects the π‘₯-axis at points 𝐴 and 𝐡, determine the area of △𝑀𝐴𝐡.

Q13:

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

  • A(π‘₯βˆ’5)+(π‘¦βˆ’4)=25
  • B(π‘₯+5)+(𝑦+4)=25
  • C(π‘₯βˆ’5)+(π‘¦βˆ’4)=5
  • D(π‘₯+5)+(𝑦+4)=5

Q14:

Given 𝐴(10,9) and 𝐡(10,βˆ’1), find the equation of the circle with diameter 𝐴𝐡.

  • Aπ‘₯+π‘¦βˆ’20π‘₯βˆ’8𝑦+91=0
  • Bπ‘₯+π‘¦βˆ’20π‘₯βˆ’8𝑦+16=0
  • Cπ‘₯+π‘¦βˆ’20π‘₯+2𝑦+76=0
  • Dπ‘₯+π‘¦βˆ’20π‘₯βˆ’18𝑦+156=0

Q15:

Write, in the form π‘Žπ‘₯+𝑏𝑦+𝑐π‘₯+𝑑𝑦+𝑒=0, the equation of the circle of radius 10 and center (βˆ’7,βˆ’8).

  • Aπ‘₯+π‘¦βˆ’14π‘₯βˆ’16𝑦+13=0
  • Bπ‘₯+𝑦+7π‘₯+8𝑦+213=0
  • Cπ‘₯+π‘¦βˆ’7π‘₯βˆ’8𝑦+213=0
  • Dπ‘₯+𝑦+14π‘₯+16𝑦+13=0

Q16:

Write, in the form π‘Žπ‘₯+𝑏𝑦+𝑐π‘₯+𝑑𝑦+𝑒=0, the equation of the circle of radius 9 and center (8,βˆ’6).

  • Aπ‘₯+𝑦+16π‘₯βˆ’12𝑦+19=0
  • Bπ‘₯+π‘¦βˆ’8π‘₯+6𝑦+181=0
  • Cπ‘₯+𝑦+8π‘₯βˆ’6𝑦+181=0
  • Dπ‘₯+π‘¦βˆ’16π‘₯+12𝑦+19=0

Q17:

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

Write the equation of the circle.

  • A(π‘₯βˆ’4)+(𝑦+8)=5
  • B(π‘₯βˆ’4)+(𝑦+8)=25
  • C(π‘₯+4)+(π‘¦βˆ’8)=5
  • D(π‘₯+4)+(π‘¦βˆ’8)=25
  • E(π‘₯βˆ’4)+(𝑦+8)=√5

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

  • A(π‘₯βˆ’10)+(𝑦+11)=75
  • B(π‘₯+2)+(𝑦+5)=75
  • C(π‘₯+2)+(𝑦+5)=15
  • D(π‘₯+2)+(𝑦+5)=225
  • E(π‘₯βˆ’10)+(𝑦+11)=225

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(π‘₯+15)+(π‘¦βˆ’14)=49
  • B(π‘₯βˆ’15)+(π‘¦βˆ’14)=49
  • C(π‘₯βˆ’15)+(𝑦+14)=49
  • D(π‘₯βˆ’15)+(𝑦+14)=196
  • E(π‘₯+15)+(π‘¦βˆ’14)=196

Q19:

Find the equation of a circle which has the radius as the circle π‘₯+𝑦+18π‘₯πœƒ+18π‘¦πœƒ+17=0cossin, and two of whose diameters lie on lines 9π‘₯+4𝑦+50=0 and r=βŸ¨βˆ’4,βˆ’1⟩+π‘ βŸ¨1,βˆ’1⟩.

  • A(π‘₯+4)+(𝑦+1)=17
  • B(π‘₯βˆ’6)+(𝑦+1)=64
  • C(π‘₯+6)+(π‘¦βˆ’1)=64
  • D(π‘₯+1)+(π‘¦βˆ’6)=17

Q20:

Determine the general form of the equation of the circle that passes through the two points 𝐴(βˆ’7,1) and 𝐡(0,6), given that the circle’s center lies on the straight line 6π‘₯βˆ’π‘¦=βˆ’43.

  • Aπ‘₯+𝑦+12π‘₯βˆ’14𝑦+48=0
  • Bπ‘₯+𝑦+14π‘₯βˆ’2𝑦+13=0
  • Cπ‘₯+π‘¦βˆ’12π‘¦βˆ’1=0
  • Dπ‘₯+π‘¦βˆ’12π‘¦βˆ’38=0

Q21:

Find the general form of the equation of the circle 𝑀 if the straight line 𝐿 of equation 2π‘₯βˆ’3𝑦=0 passes through the center of the circle and the origin.

  • Aπ‘₯+𝑦+6π‘₯+4𝑦+9=0
  • Bπ‘₯+π‘¦βˆ’4π‘₯βˆ’6𝑦+4=0
  • Cπ‘₯+𝑦+4π‘₯+6𝑦+9=0
  • Dπ‘₯+𝑦+6π‘₯+4𝑦+4=0

Q22:

A circle 𝑀 has circumference 26πœ‹ and intersects the π‘₯-axis at points (βˆ’19,0) and (5,0). What are the possible equations for 𝑀?

  • A(π‘₯+7)+(𝑦+5)=13, (π‘₯+7)+(π‘¦βˆ’5)=13
  • B(π‘₯βˆ’7)+(π‘¦βˆ’5)=169, (π‘₯βˆ’7)+(𝑦+5)=169
  • C(π‘₯βˆ’7)+(π‘¦βˆ’5)=13, (π‘₯βˆ’7)+(𝑦+5)=13
  • D(π‘₯+7)+(𝑦+5)=169, (π‘₯+7)+(π‘¦βˆ’5)=169

Q23:

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

  • Aπ‘₯+π‘¦βˆ’18π‘₯βˆ’9𝑦+81=0
  • Bπ‘₯+𝑦+18π‘₯+92𝑦+81=0
  • Cπ‘₯+𝑦+18π‘₯+9𝑦+81=0
  • Dπ‘₯+π‘¦βˆ’18π‘₯βˆ’92𝑦+81=0

Q24:

Find the center and radius of the circle π‘₯+3π‘₯+𝑦+83π‘¦βˆ’3,45536=0.

  • ACenter: ο€Ό32,43, radius: 10
  • BCenter: ο€Ό32,βˆ’43, radius: 100
  • CCenter: ο€Όβˆ’43,32, radius: 100
  • DCenter: ο€Όβˆ’32,43, radius: 100
  • ECenter: ο€Όβˆ’32,βˆ’43, radius: 10

Q25:

By completing the square, find the center and radius of the circle π‘₯+6π‘₯+π‘¦βˆ’4𝑦+8=0.

  • ACenter: (2,βˆ’3), radius: 5
  • BCenter: (βˆ’3,2), radius: 5
  • CCenter: (3,βˆ’2), radius: 5
  • DCenter: (2,βˆ’3), radius: √5
  • ECenter: (βˆ’3,2), radius: √5

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