Question

The sum of square of distances of a point from $xy-$ plane, $yz-$plane and $zx-$plane is equal to

• A. the square of its distance from origin
• B. twice the square of its distance from origin
• C. twice the sum of square of its distances from co-ordinate axes
• D. half the sum of square of its distances from co-ordinate axes

Answer 1

Answer: (A), (D)

half the sum of square of its distances from co-ordinate axes

Let point is $P(x,y,z),$ then
Distance from $xy-$plane $=|z|,$
Distance from $yz-$ plane $=|x|,$
Distance from $zx-$ plane $=|y|,$
So, sum of square of distances from planes $=x^2+y^2+z^2$
Also,
$O{P}^2=(x-0{)}^2+(y-0{)}^2+(z-0{)}^2=x^2+y^2+z^2$
(where $O$ is origin)
Now, Distance from $x-$axis $=\sqrt{y^2+z^2},$
Distance from $y-$axis $=\sqrt{x^2+z^2},$
Distance from $z-$axis $=\sqrt{y^2+x^2},$
So, sum of square of distances from co-ordinate axes
$=2(x^2+y^2+z^2)$