Length of chord of parabola ${\mathrm{y}}^2=4\mathrm{ax}$ whose equation is $\mathrm{y}-\sqrt{2}\mathrm{x}+4\sqrt{2}\mathrm{a}=0$

$\mathrm{Given} {\mathrm{y}}^2=4{\mathrm{ax}}_{ }-\left(1\right) and \mathrm{y}=\sqrt{2}x-4\sqrt{2}\mathrm{a} -\left(2\right)$

solving (1) & (2)

$$\begin{gathered} {\left(\sqrt{2}\mathrm{x}-4\sqrt{2} \mathrm{a}\right)}^2=4\mathrm{ax} \\ 2{\mathrm{x}}^2+32{\mathrm{a}}^2-16\mathrm{ax}-4\mathrm{ax}=0 \\ 2{\mathrm{x}}^2-20\mathrm{ax}+32{\mathrm{a}}^2=0 \\ {\mathrm{x}}^2-10\mathrm{ax}+16{\mathrm{a}}^2=0 \end{gathered}$$

difference  of roots $$\begin{gathered} \left|{\mathrm{x}}_1-{\mathrm{x}}_2\right|=\sqrt{100{\mathrm{a}}^2-64{\mathrm{a}}^2} \\ =6\mathrm{a} \end{gathered}$$

Length of the Chord $=\left|{\mathrm{x}}_1-{\mathrm{x}}_2\right| \sqrt{1+{\mathrm{m}}^2}$

                                     $=\left(6\mathrm{a}\right) \left(\sqrt{3}\right)=6\sqrt{3} \mathrm{a}$