3D geometry problem- 20 Nov

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)!!

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