Question

Number if sphere touching all the coordinate planes and having fixed radius:

i) 2
ii) 4
iii) 6
iv) 8​

Answer 1

Question :-

• Number if sphere touching all the coordinate planes and having fixed radius ?

Answer :-

As we know That, Their are Total 8 Octant (2 in Each Quadrant) in 3D ,

So for a fixed Radius r there will be 8 Total Number of Spheres in each Quadrant touching all the 3 Coordinate plane .

( we can see This in a 3 - D image .)

I m Uploading Image also. ( I got This from Google).

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Process :-

Now, if we Assume Radius of sphere as r , than the distance of centre from coordinates plane which touching the sphere will also be r.

So, Centre Coordinates will be = (r, r, r)

But as the centre will be in Any Quadrant , we can conclude that, its centre coordinates will be = (±r, ±r, ±r) .

And, Than,

☛ Equation of Sphere will be = (x ± r)² + (y ± r)² + (z ± r)² = r²

Or,

☛ Equation of Sphere = x² + y² + z² ± 2rx ± 2ry ± 2rz = r².

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Hence, we can Say That, If Radius is fixed Than Total 8 Sphere can be formed in the coordinate plane.

Answer 2

We have to find no. of spheres which touch all the coordinate planes in three dimensional coordinate system, formed by X, Y and Z axes.

We see the three coordinate axes make 8 divisions of the system and each division is called an octant (like quadrant in two dimensional coordinate system).

So each sphere touching all the coordinate planes are included in each octant, nearer to the origin. The planes are like tangents to the sphere.

Hence there are a total of 8 spheres touching all coordinate planes, and are,

1. The sphere in XYZ octant.

2. The sphere in XYZ' octant.

3. The sphere in XY'Z octant.

4. The sphere in XY'Z' octant.

5. The sphere in X'YZ octant.

6. The sphere in X'YZ' octant.

7. The sphere in X'Y'Z octant.

8. The sphere in X'Y'Z' octant.

Since the radius of sphere is fixed, the center of the sphere is at the same distance from the origin in each octant.