Question

If the sum of the squares of the distance of a point from the three coordinate axes be \[36\], then its distance from the origin is
A. $6$ units
B. $3\sqrt 2 $ units
C. $2\sqrt 3 $ units
D. None of these

Answer 1

Hint: Given, if the sum of the squares of the distance of a point from the three coordinate axes is \[36\]. We have to find the distance of this point from the origin. First, we will find the relation between the coordinates of point and distance using the given condition, then use that relationship to find the distance from the origin.

Formula used: If a point is given by $S(x,y,z)$, then
the sum of the squares of the distance of a point from the three coordinate axes be \[ = {\left( {\sqrt {{x^2} + {y^2}} } \right)^2} + {\left( {\sqrt {{y^2} + {z^2}} } \right)^2} + {\left( {\sqrt {{z^2} + {x^2}} } \right)^2}\]
Distance of a point from the origin\[ = \sqrt {{{(x - 0)}^2} + {{(y - 0)}^2} + {{(z - 0)}^2}} \]

Complete step by step solution:
Given, if the sum of the squares of the distance of a point from the three coordinate axes is \[36\]. Let $S(x,y,z)$ be the point.
Now, the sum of the squares of the distance of a point from the three coordinate axes be \[ = {\left( {\sqrt {{x^2} + {y^2}} } \right)^2} + {\left( {\sqrt {{y^2} + {z^2}} } \right)^2} + {\left( {\sqrt {{z^2} + {x^2}} } \right)^2}\]
\[36 = {x^2} + {y^2} + {y^2} + {z^2} + {z^2} + {x^2}\]
\[36 = 2{x^2} + 2{y^2} + 2{z^2}\]
\[36 = 2({x^2} + {y^2} + {z^2})\]
Dividing both sides by $2$
\[{x^2} + {y^2} + {z^2} = 18\]
Now, the Distance from the origin
\[ = \sqrt {{{(x - 0)}^2} + {{(y - 0)}^2} + {{(z - 0)}^2}} \]
\[ = \sqrt {{x^2} + {y^2} + {z^2}} \]
Using \[{x^2} + {y^2} + {z^2} = 18\]
$ = \sqrt {18} $
$ = 3\sqrt 2 $
Hence the distance from the origin is $3\sqrt 2 $ units

So, option (B) is the correct answer.

Note: Students should solve questions step by step to get a clarified solution. And should not misunderstand the concept of the sum of the squares of the distance of a point from the three coordinate axes. They should use the distance formula correctly to avoid mistakes while solving questions.