A relation R is defined on Z as: aRb if and only if $a^2-7ab+6b^2=0$
Then, R is: 

We have $a^2-7ab+6a^2=7a^2-7a^2=0$
Therefore, aRa for all a in Z. 
So, R is reflexive.
Let aRb, then $a^2-7ab+6b^2=0$
Consider $b^2-7ba+6a^2=b^2-\left(a^2+6b^2\right)+6a^2=5a^2-5b^2$
So, f is not symmetric.