1.4.3 * Given real numbers $a, x, y$ satisfy:
$$
x \sqrt{a(x-a)}+y \sqrt{a(y-a)}=\sqrt{|\lg (x-a)-\lg (a-y)|} .
$$

Find the value of the algebraic expression $\frac{3 x^{2}+x y-y^{2}}{x^{2}-x y+y^{2}}$.
From (1) we know $a \geqslant 0$; from (4) we know $y<a$, combining with (2) we know $a \leqslant 0$. Therefore, $a=0$, in this way, the condition equation becomes $\sqrt{|\lg x-\lg (-y)|}=0$. Thus, $x=-y, \frac{3 x^{2}+x y-y^{2}}{x^{2}-x y+y^{2}}=\frac{3 x^{2}-x^{2}-x^{2}}{x^{2}+x^{2}+x^{2}}=\frac{1}{3}$.