Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line x + 2 2 ? y = 4 If the co-ordinates of the vertex A are then the greatest integer less than or equal to is. [2026]

Question

Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line $x + 2 \sqrt{2} y = 4$ If the co-ordinates of the vertex A are $(α, β)$ then the greatest integer less than or equal to $|| α + \sqrt{2} β ||$ is. [2026]

Smart Achievers · Accepted Answer

(3)

$∵$ $m_{B C} \cdot m_{A D} = - 1$

$\Rightarrow (- \frac{1}{2 \sqrt{2}}) (\frac{β}{α}) = - 1$

$\Rightarrow β = 2 \sqrt{2} α . . . (1)$

$∵$ $O D =$ $| \frac{- 4}{\sqrt{1 + 8}} |$ $= \frac{4}{3} \Rightarrow A O = \frac{8}{3}$

$So A D = \frac{8}{3} + \frac{4}{3} = 4$

$\Rightarrow \frac{| α + 2 \sqrt{2} β - 4 |}{3} = 4 \Rightarrow α = \frac{16}{9} or - \frac{8}{9}$

${∵ A (α, β) & (0, 0) lie on same side of given line}$

$∴$ $(α, β) = (\frac{16}{9}, \frac{32 \sqrt{2}}{9}) (Rejected)$

$so (α, β) = (- \frac{8}{9}, - \frac{16 \sqrt{2}}{9})$

$= [| α + \sqrt{2} β |] = [| \frac{- 8 - 32}{9} |] = 4$

Solution

Q.
Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line $x + 2 \sqrt{2} y = 4$ If the co-ordinates of the vertex A are $(α, β)$ then the greatest integer less than or equal to $|| α + \sqrt{2} β ||$ is. [2026]

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2 5

3 4

4 3

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Solution

Q. Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line x+22y=4 If the co-ordinates of the vertex A are (α,β) then the greatest integer less than or equal to |α+2β| is. [2026]

1 2

2 5

3 4

4 3

Ans. (3) ∵ mBC·mAD=-1 ⇒(-122)(βα)=-1 ⇒β=22α ...(1) ∵ OD=|-41+8|=43⇒AO=83 So AD=83+43=4 ⇒|α+22β-4|3=4⇒α=169 or -89 {∵A(α,β) & (0,0) lie on same side of given line} ∴ (α,β)=(169,3229) (Rejected) so (α,β)=(-89,-1629) =[|α+2β|]=[|-8-329|]=4

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Q.
Let ABC be an equilateral triangle with orthocenter at the origin and the side BC on the line $x + 2 \sqrt{2} y = 4$ If the co-ordinates of the vertex A are $(α, β)$ then the greatest integer less than or equal to $|| α + \sqrt{2} β ||$ is. [2026]