\left |\begin{array}{ccc}
ax & a^2+x^2 & 1\\
ay & a^2+y^2 & 1\\
az & a^2+z^2 & 1\\
\end{array}\right |=a\left |\begin{array}{ccc}
x & a^2+x^2 & 1\\
y & a^2+y^2 & 1\\
z & a^2+z^2 & 1\\
\end{array}\right |=a\left |\begin{array}{ccc}
x-z & x^2-z^2 & 0\\
y-z & y^2-z^2 & 0\\
z & a^2+z^2 & 1\\
\end{array}\right |=a(x-z)(y-z)\left |\begin{array}{ccc}
1 & x+z & 0\\
1 & y+z & 0\\
z & a^2+z^2 & 1\\
\end{array}\right |=a(x-z)(y-z)\left |\begin{array}{cc}
1 & x+z\\
1 & y+z\\
\end{array}\right |=a(x-z)(y-z)(y-x)=a(x-y)(x-z)(z-y)