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).

  • A π‘₯ + 𝑦 + 8 π‘₯ βˆ’ 1 4 𝑦 βˆ’ 3 5 = 0  
  • B π‘₯ + 𝑦 βˆ’ 4 π‘₯ + 7 𝑦 + 1 6 5 = 0  
  • C π‘₯ + 𝑦 + 4 π‘₯ βˆ’ 7 𝑦 + 1 6 5 = 0  
  • D π‘₯ + 𝑦 βˆ’ 8 π‘₯ + 1 4 𝑦 βˆ’ 3 5 = 0  

Q2:

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

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

Q3:

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

  • A π‘₯ + 𝑦 βˆ’ 1 6 π‘₯ + 4 𝑦 βˆ’ 3 2 = 0  
  • B π‘₯ + 𝑦 + 1 6 π‘₯ βˆ’ 4 𝑦 βˆ’ 3 2 = 0  
  • C π‘₯ + 𝑦 βˆ’ 1 6 π‘₯ + 4 𝑦 + 4 3 = 0  
  • D π‘₯ + 𝑦 + 1 6 π‘₯ βˆ’ 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 𝐡 and that 𝑀𝑂=6√2.

  • A π‘₯ + 𝑦 βˆ’ 1 2 π‘₯ βˆ’ 1 2 𝑦 + 3 6 = 0  
  • B π‘₯ + 𝑦 + 1 2 π‘₯ + 1 2 𝑦 = 0  
  • C π‘₯ + 𝑦 βˆ’ 1 2 π‘₯ βˆ’ 1 2 𝑦 = 0  
  • D π‘₯ + 𝑦 βˆ’ 6 π‘₯ βˆ’ 6 𝑦 + 3 6 = 0  

Q5:

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

  • A ( π‘₯ βˆ’ 8 ) + ( 𝑦 βˆ’ 4 ) = 8 1  
  • B ( π‘₯ βˆ’ 8 ) + ( 𝑦 βˆ’ 4 ) = 9
  • C ( π‘₯ + 8 ) + ( 𝑦 + 4 ) = 9
  • D ( π‘₯ + 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,βˆ’7), and its center lies in the third quadrant.

  • A π‘₯ + ( 𝑦 + 7 ) = 2 8 9  
  • B ( π‘₯ + 1 7 ) + ( 𝑦 + 7 ) = 2 8 9  
  • C ( π‘₯ βˆ’ 1 7 ) + ( 𝑦 βˆ’ 7 ) = 2 8 9  
  • D ( π‘₯ + 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 π‘₯ + 𝑦 + 4 8 π‘₯ + 4 8 𝑦 + 5 7 6 = 0  
  • B π‘₯ + 𝑦 βˆ’ 4 8 π‘₯ βˆ’ 4 8 𝑦 + 5 7 6 = 0  
  • C π‘₯ + 𝑦 βˆ’ 4 8 π‘₯ + 4 8 𝑦 + 5 7 6 = 0  
  • D π‘₯ + 𝑦 + 2 4 π‘₯ + 2 4 𝑦 + 5 7 6 = 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 ( 1 1 , 3 )
  • B ( βˆ’ 2 , βˆ’ 1 0 )
  • C ( 2 5 , 3 4 )
  • D ( 3 , βˆ’ 9 )
  • E ( 1 0 , 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 ) + ( 𝑦 βˆ’ 7 ) = 4  
  • B ( π‘₯ βˆ’ 2 ) + ( 𝑦 βˆ’ 7 ) = 1 6  
  • C ( π‘₯ βˆ’ 2 ) + ( 𝑦 + 7 ) = 1 6  
  • D ( π‘₯ + 2 ) + ( 𝑦 βˆ’ 7 ) = 1 6  
  • 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 ) + ( 𝑦 βˆ’ 7 ) = 6 4  
  • B ( π‘₯ + 2 ) + ( 𝑦 βˆ’ 7 ) = 3 2  
  • C ( π‘₯ βˆ’ 2 ) + ( 𝑦 + 7 ) = 6 4  
  • D ( π‘₯ βˆ’ 2 ) + ( 𝑦 + 7 ) = 3 2  
  • 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 ) = 3 6  
  • C ( π‘₯ βˆ’ 2 ) + ( 𝑦 βˆ’ 5 ) = 3 6  
  • D ( π‘₯ βˆ’ 2 ) + ( 𝑦 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 ) + ( 𝑦 βˆ’ 5 ) = 4  
  • C ( π‘₯ + 2 ) + ( 𝑦 + 5 ) = 1 2  
  • D ( π‘₯ βˆ’ 2 ) + ( 𝑦 βˆ’ 5 ) = 1 2  
  • 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?

  • A π‘₯ + 𝑦 βˆ’ 1 6 π‘₯ + 4 2 𝑦 + 6 4 = 0  
  • B π‘₯ + 𝑦 βˆ’ 1 6 π‘₯ + 4 2 𝑦 + 1 2 8 = 0  
  • C π‘₯ + 𝑦 βˆ’ 1 6 π‘₯ + 4 2 𝑦 + 4 4 1 = 0  
  • D π‘₯ + 𝑦 βˆ’ 1 6 π‘₯ + 4 2 𝑦 βˆ’ 1 , 0 0 3 = 0  

Q12:

A circle of radius 15 length units has its centre 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 ) + ( 𝑦 βˆ’ 4 ) = 2 5  
  • B ( π‘₯ + 5 ) + ( 𝑦 + 4 ) = 2 5  
  • C ( π‘₯ βˆ’ 5 ) + ( 𝑦 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.

  • A π‘₯ + 𝑦 βˆ’ 2 0 π‘₯ βˆ’ 8 𝑦 + 9 1 = 0  
  • B π‘₯ + 𝑦 βˆ’ 2 0 π‘₯ βˆ’ 8 𝑦 + 1 6 = 0  
  • C π‘₯ + 𝑦 βˆ’ 2 0 π‘₯ + 2 𝑦 + 7 6 = 0  
  • D π‘₯ + 𝑦 βˆ’ 2 0 π‘₯ βˆ’ 1 8 𝑦 + 1 5 6 = 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).

  • A π‘₯ + 𝑦 βˆ’ 1 4 π‘₯ βˆ’ 1 6 𝑦 + 1 3 = 0  
  • B π‘₯ + 𝑦 + 7 π‘₯ + 8 𝑦 + 2 1 3 = 0  
  • C π‘₯ + 𝑦 βˆ’ 7 π‘₯ βˆ’ 8 𝑦 + 2 1 3 = 0  
  • D π‘₯ + 𝑦 + 1 4 π‘₯ + 1 6 𝑦 + 1 3 = 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).

  • A π‘₯ + 𝑦 + 1 6 π‘₯ βˆ’ 1 2 𝑦 + 1 9 = 0  
  • B π‘₯ + 𝑦 βˆ’ 8 π‘₯ + 6 𝑦 + 1 8 1 = 0  
  • C π‘₯ + 𝑦 + 8 π‘₯ βˆ’ 6 𝑦 + 1 8 1 = 0  
  • D π‘₯ + 𝑦 βˆ’ 1 6 π‘₯ + 1 2 𝑦 + 1 9 = 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 ) = 2 5  
  • C ( π‘₯ + 4 ) + ( 𝑦 βˆ’ 8 ) = 5  
  • D ( π‘₯ + 4 ) + ( 𝑦 βˆ’ 8 ) = 2 5  
  • 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’ 1 0 ) + ( 𝑦 + 1 1 ) = 7 5  
  • B ( π‘₯ + 2 ) + ( 𝑦 + 5 ) = 7 5  
  • C ( π‘₯ + 2 ) + ( 𝑦 + 5 ) = 1 5  
  • D ( π‘₯ + 2 ) + ( 𝑦 + 5 ) = 2 2 5  
  • E ( π‘₯ βˆ’ 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 ( π‘₯ + 1 5 ) + ( 𝑦 βˆ’ 1 4 ) = 4 9  
  • B ( π‘₯ βˆ’ 1 5 ) + ( 𝑦 βˆ’ 1 4 ) = 4 9  
  • C ( π‘₯ βˆ’ 1 5 ) + ( 𝑦 + 1 4 ) = 4 9  
  • D ( π‘₯ βˆ’ 1 5 ) + ( 𝑦 + 1 4 ) = 1 9 6  
  • E ( π‘₯ + 1 5 ) + ( 𝑦 βˆ’ 1 4 ) = 1 9 6  

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 ) = 1 7  
  • B ( π‘₯ + 6 ) + ( 𝑦 βˆ’ 1 ) = 6 4  
  • C ( π‘₯ + 1 ) + ( 𝑦 βˆ’ 6 ) = 1 7  
  • D ( π‘₯ βˆ’ 6 ) + ( 𝑦 + 1 ) = 6 4  

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.

  • A π‘₯ + 𝑦 + 1 2 π‘₯ βˆ’ 1 4 𝑦 + 4 8 = 0  
  • B π‘₯ + 𝑦 + 1 4 π‘₯ βˆ’ 2 𝑦 + 1 3 = 0  
  • C π‘₯ + 𝑦 βˆ’ 1 2 𝑦 βˆ’ 1 = 0  
  • D π‘₯ + 𝑦 βˆ’ 1 2 𝑦 βˆ’ 3 8 = 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.

  • A π‘₯ + 𝑦 + 6 π‘₯ + 4 𝑦 + 9 = 0  
  • B π‘₯ + 𝑦 βˆ’ 4 π‘₯ βˆ’ 6 𝑦 + 4 = 0  
  • C π‘₯ + 𝑦 + 4 π‘₯ + 6 𝑦 + 9 = 0  
  • D π‘₯ + 𝑦 + 6 π‘₯ + 4 𝑦 + 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 ) = 1 3   , ( π‘₯ + 7 ) + ( 𝑦 βˆ’ 5 ) = 1 3  
  • B ( π‘₯ βˆ’ 7 ) + ( 𝑦 βˆ’ 5 ) = 1 6 9   , ( π‘₯ βˆ’ 7 ) + ( 𝑦 + 5 ) = 1 6 9  
  • C ( π‘₯ βˆ’ 7 ) + ( 𝑦 βˆ’ 5 ) = 1 3   , ( π‘₯ βˆ’ 7 ) + ( 𝑦 + 5 ) = 1 3  
  • D ( π‘₯ + 7 ) + ( 𝑦 + 5 ) = 1 6 9   , ( π‘₯ + 7 ) + ( 𝑦 βˆ’ 5 ) = 1 6 9  

Q23:

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

  • A π‘₯ + 𝑦 βˆ’ 1 8 π‘₯ βˆ’ 9 𝑦 + 8 1 = 0  
  • B π‘₯ + 𝑦 + 1 8 π‘₯ + 9 2 𝑦 + 8 1 = 0  
  • C π‘₯ + 𝑦 + 1 8 π‘₯ + 9 𝑦 + 8 1 = 0  
  • D π‘₯ + 𝑦 βˆ’ 1 8 π‘₯ βˆ’ 9 2 𝑦 + 8 1 = 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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