prove that x2+4y2+16z2-2xy-8yz-4zx=0 represents a line passing through the origin!
Given expression can be written as:
\(\(\frac{x}{\sqrt 2}-y\sqrt 2\)^2+\(2\sqrt 2z-\frac{x}{\sqrt 2}\)^2+\(2\sqrt 2z-y\sqrt 2\)^2=0\).
Now, sum of squares of real numbers is zero method can be used.
If you are unable to do all the cool factorization (like me) that Manish Sir has done and need to identify the point through which the lines pass, do partial differentiation w.r.to x,y,z which will lead to three linear equations. Solving them simultaneously you will get the pt. of intersection. In your case (0,0,0). I don't know the why this method works but it is extremely effective and efficient.
If a line passes through a point ,then the values of the point satisfies the equation,in the given line if we put x=0,y=0,z=0 it satisfies the equation hence it passes through the origin.
Obvious man , put (0,0,0 & you have the answer 
to tell u mate, it was a 6 mark qn in our school's weekly test and in no way is it going to be such a cakewalk!!!! u hav got to first prove that the eqn is of a straight line which in itself is a tough qn!!! see Manish sir's ans and then u will realise its pretty tough to do, though the method of Asif Zafar is also effective!!!!!thanks Zafar!!!NIce score in limits Zafar-congrats !!!! kvs(KARTHIK V.S)