A variable plane at a distance of $$1$$ unit from the origin cuts the coordinate axes at $$A,\,B$$  and $$C.$$ If the centroid $$D\left( {x,\,y,\,z} \right)$$   satisfies the relation $${x^{ - 2}} + {y^{ - 2}} + {z^{ - 2}} = k$$     then the value of $$k$$ is :

Solution :
The plane is $$lx + my + nz = 1$$    where $${l^2} + {m^2} + {n^2} = 1.$$    It cuts axes at $$\left( {\frac{1}{l},\,0,\,0} \right),\,\left( {0,\,\frac{1}{m},\,0} \right),\,\left( {0,\,0,\,\frac{1}{n}} \right).$$
$$\therefore $$  the centroid $$ = \left( {\frac{1}{{3l}},\,\frac{1}{{3m}},\,\frac{1}{{3n}}} \right).$$     It satisfies $$\frac{1}{{{x^2}}} + \frac{1}{{{y^2}}} + \frac{1}{{{z^2}}} = k\,\,\, \Rightarrow 9\left( {{l^2} + {m^2} + {n^2}} \right) = k\,\,\, \Rightarrow 9 = k.$$