Lesson Worksheet: Equation of a Sphere Mathematics

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:

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

Q2:

Which of the following is the equation of the sphere of 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

Q3:

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

Q4:

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

Q5:

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

Q6:

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

Q7:

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)

Q8:

Give the center and radius of the sphere π‘₯+𝑦+𝑧+7π‘₯+6𝑦+3𝑧+12=0.

  • Aο€Ό72,6,32,5√102
  • Bο€Ό72,βˆ’6,32,5√102
  • Cο€Όβˆ’72,3,βˆ’32,√462
  • Dο€Όβˆ’72,βˆ’3,βˆ’32,√462

Q9:

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).

Q10:

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

This lesson includes 26 additional questions and 234 additional question variations for subscribers.

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