If 4x^2 + 9y^2 + 16z^2 -6xy - 12yz - 8zx = 0 then x,y,z are in AP , GP or HP???

multiply by 2 both sides then get an exp of this fom

(2x-3y)^2 +(3y-4z)^2+(4z-2x)^2=0

so, 2x=3y=4z=a (say)

x=a/2, y=a/3, z=a/4

so x,y,z in H.P.

this can be done by hit and trial.

put values in x,y,z as of ap,gp,hp.

the one which satisfies the equation is the solution