Parabola Ques 5
- 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} $
BETA
