A.P

Posted May 22nd, 2009 by alpha

If $x$ , $y$ and $z$ are real numbers satisfying $25(9x^2+y^2)+9z^2-15(5xy+yz+3zx)=0$ , then prove that x,y and z are in A.P

Comments

On May 22nd, 2009 Structure says:

$2(a^2+b^2+c^2-ab-ac-bc)=(a-b)^2+(b-c)^2+(a-c)^2$

a=15x
b=5y
c=3z
If

$2(a^2+b^2+c^2-ab-ac-bc)=(a-b)^2+(b-c)^2+(a-c)^2=0$

a=b=c
or 15x=5y=3z
or x=5t
y=3t
z=t
x+z=2y