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Lesson: Cartesian Equation of a Sphere in Space

In this lesson, we will learn how to find the equation of a sphere given the center and how to find the center and the radius given the sphere's equation.

Worksheet • 19 Questions

Q1:

Find the equation of a sphere that passes through the points 𝐴 ( 9 , 0 , 0 ) , 𝐡 ( 3 , 1 3 , 5 ) , and 𝐢 ( 1 1 , 0 , 1 0 ) , given that its centre lies on the 𝑦 𝑧 -plane.

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

Q2:

A sphere is tangent to the π‘₯ 𝑦 -plane, and has its centre 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 + ( 𝑧 + 3 5 ) = 1 2 2 5 2 2 2 or π‘₯ + 𝑦 + ( 𝑧 βˆ’ 3 5 ) = 1 2 2 5 2 2 2
  • B ( π‘₯ + 3 5 ) + 𝑦 + 𝑧 = 1 2 2 5 2 2 2 or ( π‘₯ βˆ’ 3 5 ) + 𝑦 + 𝑧 = 1 2 2 5 2 2 2
  • C π‘₯ + 𝑦 + 𝑧 = 1 2 2 5 2 2 2 or π‘₯ βˆ’ 𝑦 βˆ’ 𝑧 = 1 2 2 5 2 2 2
  • D π‘₯ + ( 𝑦 + 3 5 ) + 𝑧 = 1 2 2 5 2 2 2 or π‘₯ + ( 𝑦 βˆ’ 3 5 ) + 𝑧 = 1 2 2 5 2 2 2

Q3:

The spheres with equations π‘₯ + 𝑦 + 𝑧 = 9 2 2 2 and ( π‘₯ βˆ’ 4 ) + ( 𝑦 + 2 ) + ( 𝑧 βˆ’ 4 ) = 9 2 2 2 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 2 π‘₯ + 2 𝑦 + 2 𝑧 + 9 = 0
  • C 8 π‘₯ βˆ’ 4 𝑦 + 8 𝑧 βˆ’ 2 7 = 0
  • D 2 π‘₯ βˆ’ 𝑦 + 2 𝑧 + 9 = 0
  • E 8 π‘₯ βˆ’ 4 𝑦 + 8 𝑧 βˆ’ 5 4 = 0

Q4:

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

  • Ano intersection point
  • B 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 
  • C ( 3 , βˆ’ 1 , 3 )
  • D ( βˆ’ 1 , βˆ’ 2 , 3 )
  • 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 

Q5:

Determine if the given equation π‘₯ + 𝑦 βˆ’ 𝑧 + 1 2 π‘₯ + 2 𝑦 βˆ’ 4 𝑧 + 3 2 = 0 2 2 2 describes a sphere. If so, find its radius and centre.

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

Q6:

Give the equation of the sphere of centre ( 1 1 , 8 , βˆ’ 5 ) and radius 3 in standard form.

  • A ( π‘₯ βˆ’ 1 1 ) + ( 𝑦 βˆ’ 8 ) + ( 𝑧 + 5 ) = 9 2 2 2
  • B ( π‘₯ + 1 1 ) + ( 𝑦 + 8 ) + ( 𝑧 βˆ’ 5 ) = 9 2 2 2
  • C ( π‘₯ βˆ’ 1 1 ) + ( 𝑦 βˆ’ 8 ) + ( 𝑧 + 5 ) = 3 2 2 2
  • D ( π‘₯ + 1 1 ) + ( 𝑦 + 8 ) + ( 𝑧 βˆ’ 5 ) = 3 2 2 2

Q7:

The line π‘₯ + 9 βˆ’ 1 0 = 𝑦 + 4 βˆ’ 4 = 𝑧 βˆ’ 8 5 is tangent to the sphere ( π‘₯ βˆ’ 7 ) + ( 𝑦 + 3 ) + ( 𝑧 βˆ’ 7 ) = π‘Ÿ 2 2 2 2 . Find the sphere’s radius to the nearest hundredth.

Q8:

Determine if the given equation 2 π‘₯ + 2 𝑦 + 2 𝑧 + 4 π‘₯ + 4 𝑦 + 4 𝑧 βˆ’ 4 4 = 0 2 2 2 describes a sphere. If so, find its radius and centre.

  • AYes, it describes a sphere. Its radius is 5 and its centre is at ( βˆ’ 1 , βˆ’ 1 , βˆ’ 1 ) .
  • BYes, it describes a sphere. Its radius is 11 and its centre is at ( 1 , 1 , 1 ) .
  • CNo, it does not describe a sphere.
  • DYes, it describes a sphere. Its radius is 11 and its centre is at ( βˆ’ 1 , βˆ’ 1 , βˆ’ 1 ) .
  • EYes, it describes a sphere. Its radius is 5 and its centre is at ( 1 , 1 , 1 ) .

Q9:

Which of the following does the equation β€– β€– ⃑ π‘Ÿ β€– β€– βˆ’ ⃑ π‘Ÿ β‹… ο€» βˆ’ 1 0 ⃑ 𝑖 βˆ’ 6 ⃑ 𝑗 + 1 0 ⃑ π‘˜  + 5 0 = 0 2 represent?

  • Aa sphere of radius 3 length units
  • Ba plane
  • Ca circle of radius 3 length units
  • Da sphere of radius 5 √ 2 length units
  • Ea circle of radius 5 √ 2 length units

Q10:

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

Q11:

Determine if the given equation π‘₯ + 𝑦 + 𝑧 + 2 π‘₯ βˆ’ 2 𝑦 βˆ’ 8 𝑧 + 1 9 = 0 2 2 2 describes a sphere. If so, find its radius and centre.

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

Q12:

Which of the following is the equation of a sphere of the centre ( 8 , βˆ’ 1 5 , 1 0 ) and passing through ( βˆ’ 1 4 , 1 3 , βˆ’ 1 4 ) ?

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

Q13:

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

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

Q14:

Given that a sphere’s equation is ( π‘₯ + 5 ) + ( 𝑦 βˆ’ 1 2 ) + ( 𝑧 βˆ’ 2 ) βˆ’ 2 8 9 = 0 2 2 2 , determine its centre and radius.

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

Q15:

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

Q16:

Determine the surface area of the sphere of equation π‘₯ + 𝑦 + 𝑧 βˆ’ 1 4 4 4 = 0 2 2 2 , leaving your answer in terms of πœ‹ .

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

Q17:

Determine if the given equation π‘₯ + 𝑦 + 𝑧 βˆ’ 4 π‘₯ βˆ’ 6 𝑦 βˆ’ 1 0 𝑧 + 3 7 = 0 2 2 2 describes a sphere. If so, find its radius and centre.

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

Q18:

Given 𝐴 ( 0 , 4 , 4 ) , and that 𝐴 𝐡 is a diameter of the sphere ( π‘₯ + 2 ) + ( 𝑦 + 1 ) + ( 𝑧 βˆ’ 1 ) = 3 8    , what is the point 𝐡 ?

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

Q19:

Find the equation of the sphere concentric with π‘₯ + 𝑦 + 𝑧 + π‘₯ βˆ’ 5 𝑦 + 4 𝑧 = 3 2 2 2 , but with twice the radius.

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