Question

Center of sphere inscribed in a structure formed by planes x=0, y=0, z=0 and x+y+z=3+sqrt3

a)" "(0,0,0)

b)" "(1,1,0)

c)" "(1,0,1)

d)" "(1,1,1)

Solution

We can solve this using more than one method.

Method 1

Given planes x=0, y=0, z=0 and x+y+z=3+sqrt3.

Since sphere inscibed in a structure therefore radius of sphere is equal to distance between planes and center of sphere.

Let (a,b,c) is center of sphere, then

Radius = |a/1| = |b/1| = |c/1| = |(a+b+c-3-sqrt3)/sqrt3|

This equation is satisfied at a=1, b=1 and c=1.

Hence, center of sphere = (1,1,1)

Method 2

Given planes x=0, y=0, z=0 and x+y+z=3+sqrt3.

Since sphere is formed using three axes x,y,z. Hence we easily neglect (0,0,0) as it can not be the center of the sphere.

Condition of symmetry satisfied when x=y=z.

Using this concept we can neglect option b and c.

Now we are left with only one option d.

Hence, Hence, center of sphere = (1,1,1)