Question

The length of the chord of the parabola $x^2=4y$ having equation $x-\sqrt{2}y+4\sqrt{2}=0$ is :

• A. $2\sqrt{11}$
• B. $3\sqrt{2}$
• C. $6\sqrt{3}$
• D. $8\sqrt{2}$

Answer 1

Answer: (C)

$x^2=4y$
$x-\sqrt{2}y+4\sqrt{2}=0$
$x^2=4\left(\frac{x+4\sqrt{2}}{\sqrt{2}}\right)$
$\sqrt{2}{x}^2+4x+16\sqrt{2}$
$\sqrt{2}{x}^2-4x+16\sqrt{2}=0$
$x_1+x_2=2\sqrt{2};x_1{x}_2=\frac{-16\sqrt{2}}{\sqrt{2}}=-16$
${\left(\sqrt{2}y-4\sqrt{2}\right)}^2=4y$
$2y^2+32-16y=4y$
${\ell }_{AB}=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$
$=\sqrt{{\left(2\sqrt{2}\right)}^2+64+{\left(10\right)}^2-4\left(16\right)}$
$=\sqrt{8+64+100-64}$
$=\sqrt{108}=6\sqrt{3}$