The length of the chord of the parabola x ^ 2 = 4y having equation...

Solving the equations of the given parabola and the line, we get,

${\left(\sqrt{2}y - 4\sqrt{2}\right)}^{2} = 4y ⇒ {y}^{2} - 10y + 16 = 0$

$⇒ \left|{y}_{1} - {y}_{2}\right| = \sqrt{{\left({y}_{1} + {y}_{2}\right)}^{2} - 4{y}_{1}{y}_{2}} = \sqrt{100 - 64} = 6$



$⇒$ Length of the chord $= AB = \sqrt{{\left({x}_{1} - {x}_{2}\right)}^{2} + {\left({y}_{1} - {y}_{2}\right)}^{2}} = \sqrt{2{\left({y}_{1} - {y}_{2}\right)}^{2} + {\left({y}_{1} - {y}_{2}\right)}^{2}} = \sqrt{3}\left|{y}_{1} - {y}_{2}\right|$

$= 6\sqrt{3}$ units