Determinant of a particular matrix. What is the best way to find determinant of the following matrix?
$$A=\left(\begin{matrix}
1&ax&a^2+x^2\\1&ay&a^2+y^2\\ 1&az&a^2+z^2
\end{matrix}\right)$$
I thought it looks like a Vandermonde matrix, but not exactly. I can't use  $|A+B|=|A|+|B|$ to form a Vandermonde matrix. Please suggest. Thanks.

\begin{align}
&|A|\\
&=\det\left(\begin{matrix}
1&ax&a^2+x^2\\1&ay&a^2+y^2\\ 1&az&a^2+z^2
\end{matrix}\right) \\
&=\begin{vmatrix}
1&ax&a^2\\1&ay&a^2\\ 1&az&a^2
\end{vmatrix}+\begin{vmatrix}
1&ax&x^2\\1&ay&y^2\\ 1&az&z^2
\end{vmatrix} \tag{multilinearity on 3rd column} \\
&=0+a\begin{vmatrix}
1&x&x^2\\1&y&y^2\\ 1&z&z^2
\end{vmatrix} \\
&= a (x-y)(y-z)(z-x)
\end{align}
