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  • If two distinct chords are drawn from the point (4, 2) on the parabola <img alt="\mathrm{x^2=8 y}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7Bx%5E2%

If two distinct chords are drawn from the point (4, 2) on the parabola \mathrm{x^2=8 y}are bisected on the line y = mx, then the set of values of ‘m’ is given by 

 

Option: 1

\mathrm{(-\infty,-\sqrt{2}-1) \cup(\sqrt{2}-1, \infty)}


Option: 2

R


Option: 3

(0, \infty)


Option: 4

\mathrm{(-\sqrt{2}-1, \sqrt{2}-1)}


Answers (1)

best_answer

Any point on the line y = mx can be taken (t, mt).

Equation of chord of the parabola with (t, mt) as mid-point is

\mathrm{x \cdot t-4(y+m t)=t^2-8 m t}

\mathrm{\begin{aligned} & \Rightarrow 4 t-4(2+m t)=t^2-8 m t \\ & \Rightarrow t^2-4(m+1) t+8=0 \\ & D>0 \end{aligned}}

It passes through (4, 2),

For two distinct chords

\mathrm{\begin{aligned} & 16(\mathrm{~m}+1)^2-4.8>0 \\ & \mathrm{~m} \in(-\infty,-\sqrt{2}-1) \cup(\sqrt{2}-1, \infty) \end{aligned}}

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shivangi.bhatnagar

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