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Normals \mathrm{A O, A A_1, A A_2}  are drawn to parabola \mathrm{y^2=8 x} from the point \mathrm{A(h, 0)}. If triangle \mathrm{O A_1 A_2} is equilateral, then the value of \mathrm{h} is

Option: 1

28


Option: 2

18


Option: 3

25


Option: 4

23


Answers (1)

best_answer

\mathrm{Let \, A_1 \equiv\left(2 t_1^2, 4 t_1\right)}

\mathrm{ A_2 \equiv\left(2 t_1^2,-4 t_1\right) }

\mathrm{Clearly, \angle A_1 O A=\pi / 6}

\mathrm{\Rightarrow \frac{2}{t_1}=\frac{1}{\sqrt{3}} \Rightarrow t_1=2 \sqrt{3} }

\mathrm{\text{Equation of normal at }\: A_1 is}

\mathrm{ y=-t_1 x+4 t_1+2 t_1{ }^3 }

\mathrm{ \text{Since, it passes through} (h, 0) }

\mathrm{ \Rightarrow h=4+2 t_1^2=4+2 \times 4 \times 3=28 }








 

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Anam Khan

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