Equation with integers $x$, $y$ If $x$, $y$ positive integers ($x<y$), how can I solve the equation $x+y=14\sqrt{xy-48}$ ?

Write $x^2 - 194xy + y^2$ in matrix form:
$$
\pmatrix{x & y}
\pmatrix{ 1 & -97 \\ -97 & 1}
\pmatrix{x \\ y}
$$
This matrix has eigenvectors $(1,\pm 1)$, which leads us to consider $u=x+y$ and $v=x-y$.
Then $x^2 - 194xy + y^2 + 9408=0$ becomes $49 v^2 - 48 u^2+ 9408=0$.
Since $9408 =2^6×3×7^2$, this implies that $u=7u_1$ and $v=12v_1$ and so $3v_1^2-u_1^2+4=0$, a Pell equation.
