Worksheet: Equation of a Sphere

In this worksheet, we will practice finding the equation of a sphere given its center and finding the center and the radius given the sphere's equation.

Q1:

Does the given equation 2π‘₯+2𝑦+2𝑧+4π‘₯+4𝑦+4π‘§βˆ’44=0 describe a sphere? If so, find its radius and center.

  • Ayes, it describes a sphere. Its radius is 11 and its center is at (βˆ’1,βˆ’1,βˆ’1).
  • Byes, it describes a sphere. Its radius is 11 and its center is at (1,1,1).
  • Cno, it does not describe a sphere.
  • Dyes, it describes a sphere. Its radius is 5 and its center is at (1,1,1).
  • Eyes, it describes a sphere. Its radius is 5 and its center is at (βˆ’1,βˆ’1,βˆ’1).

Q2:

Determine if the given equation π‘₯+𝑦+𝑧+2π‘₯βˆ’2π‘¦βˆ’8𝑧+19=0 describes a sphere. If so, find its radius and center.

  • AYes, it describes a sphere. Its radius is 1 and its center is at (βˆ’1,1,4).
  • BYes, it describes a sphere. Its radius is 1 and its center is at (1,βˆ’1,βˆ’4).
  • CYes, it describes a sphere. Its radius is 2 and its center is at (βˆ’1,1,4).
  • DYes, it describes a sphere. Its radius is 2 and its center is at (1,βˆ’1,βˆ’4).
  • ENo, it does not describe a sphere.

Q3:

Determine if the given equation π‘₯+𝑦+π‘§βˆ’4π‘₯βˆ’6π‘¦βˆ’10𝑧+37=0 describes a sphere. If so, find its radius and center.

  • ANo, it does not describe a sphere.
  • BYes, it describes a sphere. Its radius is 1 and its center is at (βˆ’2,βˆ’3,βˆ’5).
  • CYes, it describes a sphere. Its radius is 5 and its center is at (βˆ’2,βˆ’3,βˆ’5).
  • DYes, it describes a sphere. Its radius is 5 and its center is at (2,3,5).
  • EYes, it describes a sphere. Its radius is 1 and its center is at (2,3,5).

Q4:

Find the equation of a sphere that passes through the points 𝐴(9,0,0), 𝐡(3,13,5), and 𝐢(11,0,10), given that its center lies on the 𝑦𝑧-plane.

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

Q5:

Give the equation of the sphere of center (11,8,βˆ’5) and radius 3 in standard form.

  • A ( π‘₯ + 1 1 ) + ( 𝑦 + 8 ) + ( 𝑧 βˆ’ 5 ) = 9   
  • B ( π‘₯ βˆ’ 1 1 ) + ( 𝑦 βˆ’ 8 ) + ( 𝑧 + 5 ) = 3   
  • C ( π‘₯ + 1 1 ) + ( 𝑦 + 8 ) + ( 𝑧 βˆ’ 5 ) = 3   
  • D ( π‘₯ βˆ’ 1 1 ) + ( 𝑦 βˆ’ 8 ) + ( 𝑧 + 5 ) = 9   

Q6:

Which of the following is the equation of a sphere of the center (8,βˆ’15,10) and passing through (βˆ’14,13,βˆ’14)?

  • A ( π‘₯ βˆ’ 8 ) + ( 𝑦 + 1 5 ) + ( 𝑧 βˆ’ 1 0 ) = 5 6   
  • B ( π‘₯ βˆ’ 8 ) + ( 𝑦 + 1 5 ) + ( 𝑧 βˆ’ 1 0 ) = 1 , 8 4 4   
  • C ( π‘₯ βˆ’ 8 ) βˆ’ ( 𝑦 + 1 5 ) βˆ’ ( 𝑧 βˆ’ 1 0 ) = 5 6   
  • D ( π‘₯ βˆ’ 8 ) βˆ’ ( 𝑦 + 1 5 ) βˆ’ ( 𝑧 βˆ’ 1 0 ) = 1 , 8 4 4   

Q7:

Determine the equation of a sphere with center (0,1,0), given that it touches one of the coordinate planes.

  • A π‘₯ + ( 𝑦 βˆ’ 1 ) + 𝑧 = 1   
  • B π‘₯ βˆ’ ( 𝑦 βˆ’ 1 ) βˆ’ 𝑧 = 1   
  • C π‘₯ βˆ’ ( 𝑦 + 1 ) βˆ’ 𝑧 = 1   
  • D π‘₯ + ( 𝑦 + 1 ) + 𝑧 = 1   

Q8:

Given that a sphere’s equation is (π‘₯+5)+(π‘¦βˆ’12)+(π‘§βˆ’2)βˆ’289=0, determine its center and radius.

  • A ( 5 , βˆ’ 1 2 , βˆ’ 2 ) , 289 length units
  • B ( βˆ’ 5 , 1 2 , 2 ) , 17 length units
  • C ( 5 , βˆ’ 1 2 , βˆ’ 2 ) , 17 length units
  • D ( βˆ’ 5 , 1 2 , 2 ) , 289 length units

Q9:

A sphere of radius 2 is tangent to all three coordinate planes. Given that the coordinates of its centre are all positive, what is the equation of this sphere?

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

Q10:

Determine the surface area of the sphere of equation π‘₯+𝑦+π‘§βˆ’1,444=0, leaving your answer in terms of πœ‹.

  • A 2 , 8 8 8 πœ‹
  • B 1 5 2 πœ‹
  • C 5 , 7 7 6 πœ‹
  • D 7 6 πœ‹

Q11:

Given 𝐴(0,4,4), and that 𝐴𝐡 is a diameter of the sphere (π‘₯+2)+(𝑦+1)+(π‘§βˆ’1)=38, what is the point 𝐡?

  • A ( βˆ’ 2 , βˆ’ 5 , βˆ’ 3 )
  • B ( 4 , 6 , 2 )
  • C ( βˆ’ 4 , βˆ’ 6 , βˆ’ 2 )
  • D ( 2 , 5 , 3 )

Q12:

Find the equation of the sphere concentric with π‘₯+𝑦+𝑧+π‘₯βˆ’5𝑦+4𝑧=3, but with twice the radius.

  • A ο€Ό π‘₯ + 1 2  + ο€Ό 𝑦 βˆ’ 5 2  + ( 𝑧 + 2 ) = 2 7   
  • B ο€Ό π‘₯ + 1 2  βˆ’ ο€Ό 𝑦 βˆ’ 5 2  βˆ’ ( 𝑧 + 2 ) = 2 7   
  • C ο€Ό π‘₯ + 1 2  + ο€Ό 𝑦 βˆ’ 5 2  + ( 𝑧 + 2 ) = 5 4   
  • D ο€Ό π‘₯ + 1 2  βˆ’ ο€Ό 𝑦 βˆ’ 5 2  βˆ’ ( 𝑧 + 2 ) = 5 4   

Q13:

A sphere is tangent to the π‘₯𝑦-plane, and has its center on the 𝑧-axis at a distance of 35 length units from the π‘₯𝑦-plane. What is the equation of the sphere?

  • A π‘₯ + ( 𝑦 + 3 5 ) + 𝑧 = 1 , 2 2 5    or π‘₯+(π‘¦βˆ’35)+𝑧=1,225
  • B π‘₯ + 𝑦 + 𝑧 = 1 , 2 2 5    or π‘₯βˆ’π‘¦βˆ’π‘§=1,225
  • C ( π‘₯ + 3 5 ) + 𝑦 + 𝑧 = 1 , 2 2 5    or (π‘₯βˆ’35)+𝑦+𝑧=1,225
  • D π‘₯ + 𝑦 + ( 𝑧 + 3 5 ) = 1 , 2 2 5    or π‘₯+𝑦+(π‘§βˆ’35)=1,225

Q14:

The line π‘₯+9βˆ’10=𝑦+4βˆ’4=π‘§βˆ’85 is tangent to the sphere (π‘₯βˆ’7)+(𝑦+3)+(π‘§βˆ’7)=π‘ŸοŠ¨οŠ¨οŠ¨οŠ¨. Find the sphere’s radius to the nearest hundredth.

Q15:

Which of the following does the equation ||βˆ’β‹…(βˆ’10βˆ’6+10)+50=0rrijk represent?

  • AA circle of radius 5√2 length units
  • BA plane
  • CA sphere of radius 5√2 length units
  • DA circle of radius 3 length units
  • EA sphere of radius 3 length units

Q16:

Determine if the given equation π‘₯+π‘¦βˆ’π‘§+12π‘₯+2π‘¦βˆ’4𝑧+32=0 describes a sphere. If so, find its radius and center.

  • AYes, it describes a sphere of radius 1 centered at (6,1,βˆ’2).
  • BNo, it does not describe a sphere.
  • CYes, it describes a sphere of radius 3 centered at (6,1,βˆ’2).
  • DYes, it describes a sphere of radius 1 centered at (βˆ’6,βˆ’1,2).
  • EYes, it describes a sphere of radius 2 centered at (βˆ’6,βˆ’1,2).

Q17:

The spheres with equations π‘₯+𝑦+𝑧=9 and (π‘₯βˆ’4)+(𝑦+2)+(π‘§βˆ’4)=9 intersect in a circle. Find the equation of the plane in which this circle lies.

  • A 2 π‘₯ βˆ’ 𝑦 + 2 𝑧 βˆ’ 9 = 0
  • B 8 π‘₯ βˆ’ 4 𝑦 + 8 𝑧 βˆ’ 2 7 = 0
  • C 2 π‘₯ + 2 𝑦 + 2 𝑧 + 9 = 0
  • D 8 π‘₯ βˆ’ 4 𝑦 + 8 𝑧 βˆ’ 5 4 = 0
  • E 2 π‘₯ βˆ’ 𝑦 + 2 𝑧 + 9 = 0

Q18:

Find the point(s) of intersection of the sphere (π‘₯βˆ’3)+(𝑦+1)+(π‘§βˆ’3)=9 and the line π‘₯=βˆ’1+2𝑑, 𝑦=βˆ’2βˆ’3𝑑, 𝑧=3+𝑑.

  • A ( 3 , βˆ’ 1 , 3 )
  • B ( βˆ’ 1 , βˆ’ 2 , 3 )
  • Cno intersection point
  • D ο€Ό βˆ’ 1 2 7 , βˆ’ 1 3 1 4 , 3 7 1 4  , ο€Ό βˆ’ 2 7 , βˆ’ 4 3 1 4 , 4 7 1 4 
  • E ο€Ώ βˆ’ 2 + √ 8 7 7 , βˆ’ 3 4 + 3 √ 8 7 1 4 , 4 7 + √ 8 7 1 4  , ο€Ώ βˆ’ 2 βˆ’ √ 8 7 7 , βˆ’ 3 4 βˆ’ 3 √ 8 7 1 4 , 4 7 βˆ’ √ 8 7 1 4 

Q19:

It can be shown that any four noncoplanar points determine a sphere. Find the equation of the sphere that passes through the points (0,0,0), (0,0,2), (1,βˆ’4,3) and (0,βˆ’1,3).

  • A ( π‘₯ βˆ’ 2 ) + ( 𝑦 + 2 ) + ( 𝑧 βˆ’ 1 ) = 0   
  • B ( π‘₯ βˆ’ 2 ) + ( 𝑦 + 2 ) + ( 𝑧 βˆ’ 1 ) = 9   
  • C ( π‘₯ + 2 ) + ( 𝑦 βˆ’ 2 ) + ( 𝑧 + 1 ) = 0   
  • D ( π‘₯ + 2 ) + ( 𝑦 βˆ’ 2 ) + ( 𝑧 + 1 ) = 9   
  • E π‘₯ + 𝑦 + 𝑧 = 0   

Q20:

Given that the spheres (π‘₯+3)+(𝑦+5)+(𝑧+5)=36 and (π‘₯βˆ’1)+(π‘¦βˆ’2)+(π‘§βˆ’π‘˜)=100 are tangential, determine all possible values of π‘˜.

  • A π‘˜ = βˆ’ √ 1 9 1 + 5 or π‘˜=βˆ’βˆš191βˆ’5
  • B π‘˜ = 5 + √ 1 9 1 or π‘˜=βˆ’5+√191
  • C π‘˜ = 5 + √ 1 9 1 or π‘˜=βˆ’βˆš191+5
  • D π‘˜ = βˆ’ 5 + √ 1 9 1 or π‘˜=βˆ’βˆš191βˆ’5

Q21:

Find the equation of the sphere with 𝐴=(9,βˆ’6,1) and 𝐡=(βˆ’16,βˆ’12,2) as endpoints of a diameter.

  • A ο€Ό π‘₯ + 7 2  + ( 𝑦 + 9 ) + ο€Ό 𝑧 βˆ’ 3 2  = 3 3 1 2   
  • B ο€Ό π‘₯ βˆ’ 2 5 2  + ( 𝑦 βˆ’ 3 ) + ο€Ό 𝑧 + 1 2  = 1 9 1 2   
  • C ο€Ό π‘₯ + 7 2  βˆ’ ( 𝑦 + 9 ) βˆ’ ο€Ό 𝑧 βˆ’ 3 2  = 3 3 1 2   
  • D ο€Ό π‘₯ βˆ’ 2 5 2  βˆ’ ( 𝑦 βˆ’ 3 ) βˆ’ ο€Ό 𝑧 + 1 2  = 1 9 1 2   

Q22:

A sphere of radius 50 is centered on the point on the 𝑧-axis that is at a distance 17 from the π‘₯𝑦-plane. What is the equation of the sphere?

  • A π‘₯ + 𝑦 + ( 𝑧 βˆ’ 1 7 ) = 5 0    or π‘₯+𝑦+(𝑧+17)=50
  • B π‘₯ βˆ’ 𝑦 βˆ’ ( 𝑧 βˆ’ 1 7 ) = 2 , 5 0 0    or π‘₯βˆ’π‘¦βˆ’(𝑧+17)=2,500
  • C π‘₯ + 𝑦 + ( 𝑧 βˆ’ 1 7 ) = 2 , 5 0 0    or π‘₯+𝑦+(𝑧+17)=2,500
  • D π‘₯ βˆ’ 𝑦 βˆ’ ( 𝑧 βˆ’ 1 7 ) = 5 0    or π‘₯βˆ’π‘¦βˆ’(𝑧+17)=50

Q23:

A sphere with center (π‘™βˆ’10,π‘š+4,3) and radius 2 touches the π‘₯𝑧- and 𝑦𝑧-planes. Determine all possible values of 𝑙 and π‘š.

  • A 𝑙 = 1 2 , π‘š = 6 or 𝑙=8, π‘š=2
  • B 𝑙 = βˆ’ 2 , π‘š = 1 2 or 𝑙=βˆ’6, π‘š=8
  • C 𝑙 = 1 2 , π‘š = βˆ’ 2 or 𝑙=8, π‘š=βˆ’6
  • D 𝑙 = βˆ’ 8 , π‘š = βˆ’ 2 or 𝑙=βˆ’12, π‘š=βˆ’6

Q24:

Given that 𝐴𝐡 is a diameter of a sphere whose equation is π‘₯+𝑦+𝑧+4π‘₯+5𝑦+3π‘§βˆ’18=0, and the coordinates of 𝐴 are (0,0,3), determine the coordinates of point 𝐡.

  • A ( 4 , 5 , 6 )
  • B ( βˆ’ 4 , βˆ’ 5 , βˆ’ 6 )
  • C ο€Ό 2 , 5 2 , 9 2 
  • D ο€Ό βˆ’ 2 , βˆ’ 5 2 , βˆ’ 9 2 

Q25:

Give the equation of the sphere of center (βˆ’6,15,11) that touches the π‘₯𝑦-plane.

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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.