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

Options

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

Answer

A

Explanation

Since, `(x)/(a)+(y)/(b)+(z)/(c)=1` cuts the corrdinate 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))))=1rArr(1)/(a^(2))+(1)/(b^(2))+(1)/(c^(2))=1` . . .(i)
since, P is centroid of triangle.
`:.P(x,y,z)=((a+0+0)/(3),(0+b+0)/(3),(0+0+c)/(3))`
`rArr" "x=(a)/(3),y=(b)/(3),z=(c)/(3)` . . . (ii)
From Eqs. (i) and (ii), we get
`(1)/(9x^(2))+(1)/(9y^(2))+(1)/(9z^(2))=1`
`rArr" "(1)/(x^(2))+(1)/(y^(2))+(1)/(z^(2))=9`
`:." "=9`