# 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 describe a sphere? If so, find its radius and center.

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

Q2:

Determine if the given equation describes a sphere. If so, find its radius and center.

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

Q3:

Determine if the given equation 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 .
• CYes, it describes a sphere. Its radius is 5 and its center is at .
• DYes, it describes a sphere. Its radius is 5 and its center is at .
• EYes, it describes a sphere. Its radius is 1 and its center is at .

Q4:

Find the equation of a sphere that passes through the points , , and , given that its center lies on the -plane.

• A
• B
• C
• D

Q5:

Give the equation of the sphere of center and radius 3 in standard form.

• A
• B
• C
• D

Q6:

Which of the following is the equation of a sphere of the center and passing through ?

• A
• B
• C
• D

Q7:

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

• A
• B
• C
• D

Q8:

Given that a sphere’s equation is , determine its center and radius.

• A , 289 length units
• B , 17 length units
• C , 17 length units
• D , 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
• B
• C
• D

Q10:

Determine the surface area of the sphere of equation , leaving your answer in terms of .

• A
• B
• C
• D

Q11:

Given , and that is a diameter of the sphere , what is the point ?

• A
• B
• C
• D

Q12:

Find the equation of the sphere concentric with , but with twice the radius.

• A
• B
• C
• D

Q13:

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

• A or
• B or
• C or
• D or

Q14:

The line is tangent to the sphere . Find the sphere’s radius to the nearest hundredth.

Q15:

Which of the following does the equation represent?

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

Q16:

Determine if the given equation describes a sphere. If so, find its radius and center.

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

Q17:

The spheres with equations and intersect in a circle. Find the equation of the plane in which this circle lies.

• A
• B
• C
• D
• E

Q18:

Find the point(s) of intersection of the sphere and the line , , .

• A
• B
• Cno intersection point
• D ,
• E ,

Q19:

It can be shown that any four noncoplanar points determine a sphere. Find the equation of the sphere that passes through the points , , and .

• A
• B
• C
• D
• E

Q20:

Given that the spheres and are tangential, determine all possible values of .

• A or
• B or
• C or
• D or

Q21:

Find the equation of the sphere with and as endpoints of a diameter.

• A
• B
• C
• D

Q22:

A sphere of radius 50 is centered on the point on the -axis that is at a distance 17 from the -plane. What is the equation of the sphere?

• A or
• B or
• C or
• D or

Q23:

A sphere with center and radius 2 touches the - and -planes. Determine all possible values of and .

• A , or ,
• B , or ,
• C , or ,
• D , or ,

Q24:

Given that is a diameter of a sphere whose equation is , and the coordinates of are , determine the coordinates of point .

• A
• B
• C
• D

Q25:

Give the equation of the sphere of center that touches the -plane.

• A
• B
• C
• D