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

**Q1: **

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

- AYes, it describes a sphere. Its radius is 5 and its centre is at .
- BNo, it does not describe a sphere.
- CYes, it describes a sphere. Its radius is 11 and its centre is at .
- DYes, it describes a sphere. Its radius is 5 and its centre is at .
- EYes, it describes a sphere. Its radius is 11 and its centre is at .

**Q2: **

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

- AYes, it describes a sphere. Its radius is 1 and its centre is at .
- BYes, it describes a sphere. Its radius is 1 and its centre is at .
- CYes, it describes a sphere. Its radius is 2 and its centre is at .
- DNo, it does not describe a sphere.
- EYes, it describes a sphere. Its radius is 2 and its centre is at .

**Q3: **

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

- AYes, it describes a sphere. Its radius is 1 and its centre is at .
- BNo, it does not describe a sphere.
- CYes, it describes a sphere. Its radius is 5 and its centre is at .
- DYes, it describes a sphere. Its radius is 1 and its centre is at .
- EYes, it describes a sphere. Its radius is 5 and its centre is at .

**Q4: **

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

- A
- B
- C
- D

**Q6: **

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

- A
- B
- C
- D

**Q7: **

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

- A
- B
- C
- D

**Q8: **

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

- A , 289 length units
- B , 289 length units
- C , 17 length units
- D , 17 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 centre 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 circle of radius 3 length units
- CA sphere of radius length units
- DA sphere of radius 3 length units
- EA plane

**Q16: **

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

- AYes, it describes a sphere of radius 1 centred at .
- BYes, it describes a sphere of radius 1 centred at .
- CYes, it describes a sphere of radius 3 centred at .
- DNo, it does not describe a sphere.
- EYes, it describes a sphere of radius 2 centred 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
- C
- Dno intersection point
- 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