Sphere with radius 7 tangent to xy-plane with center on z-axis

[MidgetDwarf]

Write the equation of the sphere with radius 7 and center on the positive z-axis, if the sphere is tangent to the plane z=0.

I know this is an easy problem if i understood the terminology better.

The equation of a sphere (x-h)^2 +(y-k)^2 +(z-l)^2 = 49.

I know that the plane z is the set S={(x,y,0)| z=0}, it is a plane parallel to the xy-axis (vertical plane).

What does it mean by, " with radius 7 and center on the positive z-axis"?

Is it x^2+y^2 +(z-l)^2=49 ?

I am not sure how to proceed.

At first i used the equation of the sphere.

(x-h)^2 +(y-k)^2 +(z-l)^2 = 49.

and I let h=x k=y and z=0 (points taken from the plane)

I end up with (-l)^2=49

l=7.

I need a new equation representing the the tangent from the sphere and plane.

can I say x^2 +y^2 +(z-7)^2= 49? It is the answer in the back of the book.

My argument looks very weak and flawed.

 

[SteamKing]

Write the equation of the sphere with radius 7 and center on the positive z-axis, if the sphere is tangent to the plane z=0.

I know this is an easy problem if i understood the terminology better.

The equation of a sphere (x-h)^2 +(y-k)^2 +(z-l)^2 = 49.

I know that the plane z is the set S={(x,y,0)| z=0}, it is a plane parallel to the xy-axis (vertical plane).

What does it mean by, " with radius 7 and center on the positive z-axis"?

The radius of the sphere = 7. What's unclear about that?

If the center of the sphere is located somewhere on the positive z-axis, what does this tell you about the x and y coordinates of the center? Where is the z-axis located in the xy-plane?

If the sphere:

1. has a radius of 7 and
2. the center is located on the positive z-axis and
3. is tangent to the plane z = 0

Can't you use the geometry of the sphere to determine the coordinates of the center? This can be done by inspection (i.e., without the need for calculation).

 

[C. Lee]

Write the equation of the sphere with radius 7 and center on the positive z-axis, if the sphere is tangent to the plane z=0.

Remind that a plane tangent to a sphere is always vertical to the radius joining the center of the sphere and the point where they meet.

Also what does it mean for the sphere to have its center on the positive z-axis, which is a part of x=y=0?

 

[SteamKing]

Problem needs no calculus to be solved. Moved to the Pre-calculus HW forum (again).

 

[MidgetDwarf]

Remind that a plane tangent to a sphere is always vertical to the radius joining the center of the sphere and the point where they meet.

Also what does it mean for the sphere to have its center on the positive z-axis, which is a part of x=y=0?

It means that it's center is (0,0,l), we don't know what point l is, only that l is in the positive direction of the z-axis. The z plane is a horizontal plane. so i measure from the center to the z plane. I get X^2+ Y^2 + (Z-7)^2 =49.

 

[C. Lee]

It means that it's center is (0,0,l), we don't know what point l is, only that l is in the positive direction of the z-axis. The z plane is a horizontal plane. so i measure from the center to the z plane. I get X^2+ Y^2 + (Z-7)^2 =49.

Correct.

 

[SteamKing]

It means that it's center is (0,0,l), we don't know what point l is, only that l is in the positive direction of the z-axis. The z plane is a horizontal plane. so i measure from the center to the z plane. I get X^2+ Y^2 + (Z-7)^2 =49.

You're not thinking thru this clearly. Making a sketch may help.

If the plane z = 0 is tangent to the sphere, how far away is it from the center of the sphere?

 

[HallsofIvy]

It means that it's center is (0,0,l), we don't know what point l is, only that l is in the positive direction of the z-axis.

Badly worded. "l" is not a point, it is a number. And since it is a number, l is not in any direction. If the xy-plane is tangent to the sphere, what is l?

The z plane is a horizontal plane. so i measure from the center to the z plane.

There is NO "z plane". You mean either the "z= 0 plane" or the "xy- plane".

I get X^2+ Y^2 + (Z-7)^2 =49.

Yes, that is correct. Now, how did you get the "Z- 7"?[/quote][/QUOTE]