Official solution
x2 = 4y
x - \[\sqrt 2 \]y + 4\[\sqrt 2 \] = 0
Solving together we get
x2 = 4\[\left( {{{x + 4\sqrt 2 } \over {\sqrt 2 }}} \right)\]
\[\sqrt 2 \]x2 + 4x + 16\[\sqrt 2 \]
\[\sqrt 2 \]x2 - 4x - 16\[\sqrt 2 \] = 0
x1 + x2 = 2\[\sqrt 2 \]; x1x2 = \[{{ - 16\sqrt 2 } \over {\sqrt 2 }}\] = - 16
Similarly,
(\[\sqrt 2 \]y - 4\[\sqrt 2 \])2 = 4y
2y2 + 32 - 16y = 4y
\[\ell \]AB = \[\sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} \]
\[ = \sqrt {{{\left( {2\sqrt 2 } \right)}^2} + 64 + {{\left( {10} \right)}^2} - 4\left( {16} \right)} \]
\[ = \sqrt {8 + 64 + 100 - 64} \]
\[ = \sqrt {108} = 6\sqrt 3 \]