Parabola Ques 5

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

(2019 Main, 10 Jan II)

(a) $8 \sqrt{2}$

(b) $2 \sqrt{11}$

(c) $3 \sqrt{2}$

(d) $6 \sqrt{3}$

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Answer:

Correct Answer: 5.(d)

Solution: (d) Given, equation of parabola is $x^2=4 y \quad ……..(i)$

and the chord is $x-\sqrt{2} y+4 \sqrt{2}=0 \quad ……..(ii)$

From Eqs. (i) and (ii), we have

$ {[\sqrt{2}(y-4)]^2=4 y} $

$\Rightarrow \quad 2(y-4)^2=4 y $

$\Rightarrow \quad (y-4)^2=2 y $

$\Rightarrow \quad y^2-8 y+16=2 y $

$\Rightarrow \quad y^2-10 y+16=0 \quad ……..(iii)$

Let the roots of Eq. (iii) be $y_1$ and $y_2$

Then,

$ y_1+y_2=10 \text { and } y_1 y_2=16 \quad ……..(iv)$

Again from Eqs. (i) and (ii), we have

$ x^2=4\left[\frac{x}{\sqrt{2}}+4\right] $

$\Rightarrow \quad x^2-2 \sqrt{2} x-16=0 \quad ……..(v)$

Let the roots of $\mathrm{Eq}$. $(v)$ be $x_1$ and $x_2$

Then,

$x_1+x_2 =2 \sqrt{2} $

$x_1 x_2 =-16 \quad ……..(vi)$

Clearly, length of the chord $A B$

$ \begin{aligned} & =\sqrt{\left(x_1-x_2\right)^2+\left(y_1-y_2\right)^2} \\ & =\sqrt{\left(x_1+x_2\right)^2-4 x_1 x_2+\left(y_1+y_2\right)^2-4 y_1 y_2} \quad \quad \left[\because(a-b)^2=(a+b)^2-4 a b\right] \\ & =\sqrt{8+64+100-64} \\ & =\sqrt{108}=6 \sqrt{3} \quad \quad \text { [from Eqs. (iv) and (vi)] } \end{aligned} $



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