First slide

Parabola

Question

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

Moderate

A

$2 \sqrt{11} a$
B

$4 \sqrt{2} a$
C

$8 \sqrt{2} a$
D

$6 \sqrt{3} a$

Solution

Any point on the parabola is $(a t^{2}, 2 a t)$ ;
$\begin{array}{l} 2 a t - a t^{2} \sqrt{2} + 4 a \sqrt{2} = 0 \\ \Rightarrow \sqrt{2} t^{2} - 2 t - 4 \sqrt{2} = 0 \\ \Rightarrow t_{1} + t_{2} = \sqrt{2}, t_{1} t_{2} = - 4 \\ \Rightarrow t_{1} - t_{2} = \sqrt{18} \end{array}$
$\sqrt{{(a t_{1}^{2} - a t_{2}^{2})}^{2} + {(2 a t_{1} - 2 a t_{2})}^{2}} = a \sqrt{18} \sqrt{2 + 4} = 6 \sqrt{3} a$ .

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