Question

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

Answer 1

Correct Answer - A
Since, `(x)/(a)+(y)/(b)+(z)/( c )=1` cuts the coordinate axes at
`A(a,0,0),B(0,b, 0),C(0,0,c).`
And its distance from origin=1

`:." "(1)/(sqrt((1)/(a^(2))+(1)/(b^(2))+(1)/(c^(2))))=1`
or`" "(1)/(a^(2))+(1)/(b^(2))+(1)/(c^(2))=1" "...(i)`
where, P is centroid of triangle.
`:." "P(x,y,z)=((a+0+0)/(3),(0+b+0)/(3),(0+0+c)/(3))`
`implies" "x=(a)/(3),y=(b)/(3),z=(c)/(3)" "...(ii)`
From Eqs. (i) and (ii),
`(1)/(9x^(2))+(1)/(9y^(2))+(1)/(9z^(2))=1`
or`" "(1)/(x^(2))+(1)/(y^(2))+(1)/(z^(2))=9=K`
`:." "K=9`