A variable plane at distance of `1` unit from the origin cuts the coordinte axes at `A,B` and `C`. If the centroid `D(x,y,z)` of triangle `ABC` satisfies the relation `1/x^2+1/y^2+1/z^2=k` then the value of `k` is

Options

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

Answer

d

Explanation

Let the equation of the variable plane be `(x)/(a)+ (y)/(b) + (z)/(c )=1`, which meets the axes at `A(a, 0, 0), B(0, b, 0) and C(0, 0, c)`.
The centroid of `DeltaABC` is `((a)/(3), (b)/(3), (c)/(3))` and it satisfies the relation `(1)/(x^(2))+(1)/(y^(2))+(1)/(z^(2))=k`. Thus,
`" "(9)/(a^(2))+ (9)/(b^(2))+ (9)/(c^(2))=k`
or `" "(1)/(a^(2))+ (1)/(b^(2))+ (1)/(c^(2))=(k)/(9)" "` (i)
Also it is given that the distance of the plane `(x)/(a)+ (y)/(b)+ (z)/(c)=1` from (0, 0, 0) is 1 unit. Therefore,
`" "(1)/(sqrt((1)/(a^(2))+(1)/(b^(2))+ (1)/(c^(2))))=1 or (1)/(a^(2))+ (1)/(b^(2))+ (1)/(c^(2))=1" "`(ii)
From (i) and (ii), we get `k//9=1` , i.e., `k=9`