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Question

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


A
211
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B
32
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C
63
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D
82
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Solution

The correct option is D $$6\sqrt{3}$$
$$x^2 = 4y$$
$$x - \sqrt{2} y  + 4\sqrt{2} = 0$$

Solving together we get
$$x^2 = 4 \left(\dfrac{x + 4\sqrt{2}}{\sqrt{2}}\right)$$
$$\sqrt{2}x^2 - 4x - 16\sqrt{2}=0$$
$$x_1 + x_2 = 2\sqrt{2}$$;  $$x_1x_2 = \dfrac{-16\sqrt{2}}{\sqrt{2}}=-16$$

Similarly,
$$(\sqrt{2} y - 4\sqrt{2})^2 = 4y$$
$$2y^2 + 32 - 16y = 4y$$
$$ 2y^2 -20y + 32 = 0 < ^{y_1 + y_2 = 10}_{y_1y_2=16}$$

$$\ell_{AB} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$$ 
$$= \sqrt{(2\sqrt{2})^2 + 64 + (10)^2 - 4(16)}$$

$$=\sqrt{8 + 64 + 100 - 64}$$
$$= \sqrt{108} = 6\sqrt{3}$$

1143144_1332497_ans_b3ab5063d48a4084b352b8a28ced9c24.png

Mathematics

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