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Q. If the distance of the point $ (1, 1, 1) $ from the origin is half its distance from the plane $ x+y+z+k=0, $ then $ k $ is equal to:

KEAMKEAM 2005

Solution:

Distance of a point (1, 1, 1) from $ x+y+z+k=0 $
is $ \left| \frac{1+1+1+k}{\sqrt{3}} \right| $
According to question, $ \left| \frac{3+k}{\sqrt{3}} \right|= $
$ \pm 2\sqrt{3} $ $ \Rightarrow $ $ k=3,-9 $