Making the condition $a^{\,2}+ b^{\,2}+ c^{\,2}+ k\,abc= k+ 3$ homogeneous

Question

I saw it on AoPS, very great: $\lceil$ https://artofproblemsolving.com/community/c6h1274759p6726915 $\rfloor$
JunBo-Yang used his own substitution for the condition: $a^{\,2}+ b^{\,2}+ c^{\,2}+ 3\,abc= 6$ , then:
$$a= \frac{\left ( 3- \sqrt{3} \right )x+ \left ( 3+ \sqrt{3} \right )y}{\sqrt{\left ( 4\,y+ z+ x \right )\left ( 4\,x+ y+ z \right )}}$$
$$b= \frac{\left ( 3- \sqrt{3} \right )z+ \left ( 3+ \sqrt{3} \right )x}{\sqrt{\left ( 4\,x+ y+ z \right )\left ( 4\,z+ x+ y \right )}}$$
$$c= \frac{\left ( 3- \sqrt{3} \right )y+ \left ( 3+ \sqrt{3} \right )z}{\sqrt{\left ( 4\,z+ x+ y \right )\left ( 4\,y+ z+ x \right )}}$$
By the same way, the condition: $a^{\,2}+ b^{\,2}+ c^{\,2}+ k\,abc= k+ 3$ can be substituted by:
$$a= \frac{\left ( k- \sqrt{k} \right )x+ \left ( k+ \sqrt{k} \right )y}{\sqrt{\left ( \left ( k+ 1 \right )\,y+ z+ x \right )\left ( \left ( k+ 1 \right )\,x+ y+ z \right )}}$$
$$b= \frac{\left ( k- \sqrt{k} \right )z+ \left ( k+ \sqrt{k} \right )x}{\sqrt{\left ( \left ( k+ 1 \right )\,x+ y+ z \right )\left ( \left ( k+ 1 \right )\,z+ x+ y \right )}}$$
$$c= \frac{\left ( k- \sqrt{k} \right )y+ \left ( k+ \sqrt{k} \right )z}{\sqrt{\left ( \left ( k+ 1 \right )\,z+ x+ y \right )\left ( \left ( k+ 1 \right )\,y+ z+ x \right )}}$$

Answer 1

If your substitution is true it should be true for $k=0$, but it's not so.