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 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.

  • Aπ‘₯+𝑦+π‘§βˆ’2π‘¦βˆ’7π‘§βˆ’81=0
  • Bπ‘₯βˆ’π‘¦βˆ’π‘§+4𝑦+14𝑧+81=0
  • Cπ‘₯+𝑦+π‘§βˆ’4π‘¦βˆ’14π‘§βˆ’81=0
  • Dπ‘₯βˆ’π‘¦βˆ’π‘§+2𝑦+7𝑧+81=0

Q5:

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

  • A(π‘₯+11)+(𝑦+8)+(π‘§βˆ’5)=9
  • B(π‘₯βˆ’11)+(π‘¦βˆ’8)+(𝑧+5)=3
  • C(π‘₯+11)+(𝑦+8)+(π‘§βˆ’5)=3
  • D(π‘₯βˆ’11)+(π‘¦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+15)+(π‘§βˆ’10)=56
  • B(π‘₯βˆ’8)+(𝑦+15)+(π‘§βˆ’10)=1,844
  • C(π‘₯βˆ’8)βˆ’(𝑦+15)βˆ’(π‘§βˆ’10)=56
  • D(π‘₯βˆ’8)βˆ’(𝑦+15)βˆ’(π‘§βˆ’10)=1,844

Q7:

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

  • Aπ‘₯+(𝑦+1)+𝑧=1
  • Bπ‘₯+(π‘¦βˆ’1)+𝑧=1
  • Cπ‘₯βˆ’(𝑦+1)βˆ’π‘§=1
  • Dπ‘₯βˆ’(π‘¦βˆ’1)βˆ’π‘§=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’12,βˆ’2), 289 length units
  • B(βˆ’5,12,2), 17 length units
  • C(5,βˆ’12,βˆ’2), 17 length units
  • D(βˆ’5,12,2), 289 length units

Q9:

A sphere of radius 2 is tangent to all three ordered pairs. Given that the ordered pairs of its center are all positive, what is the equation of this sphere?

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

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‹.

  • A2,888πœ‹
  • B152πœ‹
  • C5,776πœ‹
  • D76π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.

  • Aο€Όπ‘₯+12+ο€Όπ‘¦βˆ’52+(𝑧+2)=27
  • Bο€Όπ‘₯+12οˆβˆ’ο€Όπ‘¦βˆ’52οˆβˆ’(𝑧+2)=27
  • Cο€Όπ‘₯+12+ο€Όπ‘¦βˆ’52+(𝑧+2)=54
  • Dο€Όπ‘₯+12οˆβˆ’ο€Όπ‘¦βˆ’52οˆβˆ’(𝑧+2)=54

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?

  • Aπ‘₯+(𝑦+35)+𝑧=1,225 or π‘₯+(π‘¦βˆ’35)+𝑧=1,225
  • Bπ‘₯+𝑦+𝑧=1,225 or π‘₯βˆ’π‘¦βˆ’π‘§=1,225
  • C(π‘₯+35)+𝑦+𝑧=1,225 or (π‘₯βˆ’35)+𝑦+𝑧=1,225
  • Dπ‘₯+𝑦+(𝑧+35)=1,225 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:

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).

Q16:

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.

  • A2π‘₯βˆ’π‘¦+2π‘§βˆ’9=0
  • B8π‘₯βˆ’4𝑦+8π‘§βˆ’54=0
  • C2π‘₯βˆ’π‘¦+2𝑧+9=0
  • D2π‘₯+2𝑦+2𝑧+9=0
  • E8π‘₯βˆ’4𝑦+8π‘§βˆ’27=0

Q17:

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
  • Dο€Όβˆ’127,βˆ’1314,3714, ο€Όβˆ’27,βˆ’4314,4714
  • Eο€Ώβˆ’2+√877,βˆ’34+3√8714,47+√8714, ο€Ώβˆ’2βˆ’βˆš877,βˆ’34βˆ’3√8714,47βˆ’βˆš8714

Q18:

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

Q19:

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‘˜.

  • Aπ‘˜=βˆ’βˆš191+5 or π‘˜=βˆ’βˆš191βˆ’5
  • Bπ‘˜=5+√191 or π‘˜=βˆ’5+√191
  • Cπ‘˜=5+√191 or π‘˜=βˆ’βˆš191+5
  • Dπ‘˜=βˆ’5+√191 or π‘˜=βˆ’βˆš191βˆ’5

Q20:

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

  • Aο€Όπ‘₯+72+(𝑦+9)+ο€Όπ‘§βˆ’32=3312
  • Bο€Όπ‘₯βˆ’252+(π‘¦βˆ’3)+𝑧+12=1912
  • Cο€Όπ‘₯+72οˆβˆ’(𝑦+9)βˆ’ο€Όπ‘§βˆ’32=3312
  • Dο€Όπ‘₯βˆ’252οˆβˆ’(π‘¦βˆ’3)βˆ’ο€Όπ‘§+12=1912

Q21:

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?

  • Aπ‘₯+𝑦+(π‘§βˆ’17)=50 or π‘₯+𝑦+(𝑧+17)=50
  • Bπ‘₯βˆ’π‘¦βˆ’(π‘§βˆ’17)=2,500 or π‘₯βˆ’π‘¦βˆ’(𝑧+17)=2,500
  • Cπ‘₯+𝑦+(π‘§βˆ’17)=2,500 or π‘₯+𝑦+(𝑧+17)=2,500
  • Dπ‘₯βˆ’π‘¦βˆ’(π‘§βˆ’17)=50 or π‘₯βˆ’π‘¦βˆ’(𝑧+17)=50

Q22:

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.

  • A𝑙=12, π‘š=6 or 𝑙=8, π‘š=2
  • B𝑙=βˆ’2, π‘š=12 or 𝑙=βˆ’6, π‘š=8
  • C𝑙=12, π‘š=βˆ’2 or 𝑙=8, π‘š=βˆ’6
  • D𝑙=βˆ’8, π‘š=βˆ’2 or 𝑙=βˆ’12, π‘š=βˆ’6

Q23:

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)
  • Cο€Ό2,52,92
  • Dο€Όβˆ’2,βˆ’52,βˆ’92

Q24:

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

  • A(π‘₯+6)+(π‘¦βˆ’15)+(π‘§βˆ’11)=121
  • B(π‘₯+6)+(π‘¦βˆ’15)+(π‘§βˆ’11)=11
  • C(π‘₯+6)βˆ’(π‘¦βˆ’15)βˆ’(π‘§βˆ’11)=121
  • D(π‘₯+6)βˆ’(π‘¦βˆ’15)βˆ’(π‘§βˆ’11)=11

Q25:

Find the equation of a sphere that passes through the points 𝐴(0,3,βˆ’2) and 𝐡(βˆ’1,βˆ’3,βˆ’5), given that its center lies on the 𝑧-axis.

  • Aπ‘₯+𝑦+𝑧+113=1069
  • Bπ‘₯+𝑦+ο€Όπ‘§βˆ’113=1069
  • Cπ‘₯βˆ’π‘¦βˆ’ο€Όπ‘§βˆ’113=1069
  • Dπ‘₯βˆ’π‘¦βˆ’ο€Όπ‘§+113=1069

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