Q. The length of the chord of the parabola $x^2 = 4y$ having equation $x - \sqrt{2} y + 4 \sqrt{2} = 0$ is :
Solution:
$ x^{2} =4y $
$ x-\sqrt{2} y+4\sqrt{2} =0 $
$ x^{2} =4 \left(\frac{x+4\sqrt{2}}{\sqrt{2}}\right) $
$ \sqrt{2}x^{2} +4x+16\sqrt{2} $
$ \sqrt{2}x^{2} - 4x+16\sqrt{2} = 0 $
$x_{1}+ x_{2} =2 \sqrt{2} ; x_{1}x_{2} = \frac{-16\sqrt{2}}{\sqrt{2}} = -16 $
$\left(\sqrt{2}y -4\sqrt{2}\right)^{2} =4y $
$2y^{2} +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} $
