A variable plane `x/a+y/b+z/c=1` at a unit distance from origin cuts the coordinate axes at `A, B and C.` Centroid `(x, y, z)` satisfies the equation `1/x^2+1/y^2+1/z^2=K.` The value of `K` is (A) `9` (B) `3` (C) `1/9` (D) `1/3`

Options

A. 9
B. 3
C. `1//9`
D. `1//3`

Answer

A

Explanation

Given plane cuts the coordinate axes at `A(a,0,0),B(0,b,0)` and `C(0,0,0)`. It is at a unit distance from the origin.
`:.1/(sqrt(1/(a^(2))+1/(b^(2))+1/(c^(2))))=1implies1/(a^(2))+1/(b^(2))+1/(c^(2))=1`……………i
Since `(x,y,z)` is the centroid of `DeltaABC`.
`:.x=a/3,y=b/3,z=c/3impliesa=3x,b=3y,c=3z`
Substituting the values of `a,b,c` in (i) we get
`1/(x^(2))+1/(y^(2))+1/(z^(2))=9impliesk=9`