For $(b)$ simply take $(a,b,c)=(1,1,1)$, so that $ E(x,y,z):=xa+yb+zc $ vanishes for all $(x,y,z)=(x,-x,0)$, but is always an integer. For $(a)$ consider $(x_n,y_n)$ to be all the positive solutions of the Pell's Equation $3x^2-2y^2=1$. Then: \[3x^2-2y^2=1\iff (x\sqrt3-y\sqrt2)=\frac{1}{x\sqrt3+y\sqrt2}\]So $\frac{1}{x_n\sqrt3+y_n\sqrt2}\in\{ |x\sqrt 2+y\sqrt 3+y\sqrt 5|\big| x,y,z\in\mathbb{Z} ,x^2+y^2+z^2>0 \}$, but as $n$ goes to infinity, $\frac{1}{x_n\sqrt3+y_n\sqrt2}$ goes to 0. NOTE:I'm right now to lazy to show why this equation has infinite integer solutions, but if necessary I might try to sketch a proof.