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  • The length of the chord of the parabola   having equation <img alt="x-\sqrt{2} y+4

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

Option: 1

\begin{aligned} & 6 \sqrt{3} \\ \end{aligned}


Option: 2

8 \sqrt{3} \\


Option: 3

-6 \sqrt{3} \\


Option: 4

-8 \sqrt{3}


Answers (1)

best_answer

Given, equation of parabola is  x^2=4 y —--------------------------(i)

and the chord is  x-\sqrt{2} y+4 \sqrt{2}=0 —-----------------------(ii)

From (i) and (ii),

\begin{aligned} & {[\sqrt{2}(y-4)]^2=4 y } \\ \Rightarrow & 2(y-4)^2=4 y \\ \Rightarrow & (y-4)^2=2 y \\ \Rightarrow & y^2-8 y+16=2 y \\ \Rightarrow & y^2-10 y+16=0 \end{aligned}..........................(iii)

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

Now, y_1+y_2=10 \text { and } y_1 \cdot y_2=16 —---------(iv)

from(i) and (ii)

\begin{aligned} & x^2=4\left[\frac{x}{\sqrt{2}}+4\right] \\ & \Rightarrow x^2-2 \sqrt{2} x-16=0 \end{aligned}..............................(v)

Let the roots of Eq. (v) be x_1 and x_2

 length of the chord AB = 

\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} \\ & =\sqrt{8+64+100-64} \\ & =\sqrt{108}=6 \sqrt{3} \end{aligned}

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Divya Prakash Singh

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