Question

Find the equation of a sphere, whose centre is $(1,1,1)$ and radius is $5$ units

• A. $x^2+y^2+z^2-x-y-z=5$
• B. $x^2+y^2+z^2-xx-2y-2z=5$
• C. $x^2+y^2+z^2-x-y-z=25$
• D. $x^2+y^2+z^2-2x-2y-2z=22$

Answer 1

The correct option is D

$x^2+y^2+z^2-2x-2y-2z=22$

Sphere can be thought of as a three dimensional extension of a circle. Similar to distance formula and section formula, the equation of sphere has additional terms involving z coordinate. To find the equation of the sphere, let’s consider a general point $P(x,y,z)$ on the sphere. Let the centre of the sphere be $O(1,1,1)$. We are given that the radius is $5$ units.

Distance of P from O will be equal to the radius, that is 5 units. Using distance formula, we get

$PO=\sqrt{(x-1{)}^2+(y-1{)}^2+(z-1{)}^2}=5\Rightarrow (x-1{)}^2+(y-1{)}^2+(z-1{)}^2=25\Rightarrow x^2+y^2+z^2-2x-2y-2z=22$
This is the equation of the sphere with centre $(1,1,1)$ and radius $5$ units