If z 1- z 2- z 3 represent the vertices of an equilateral - ABC and - z 2- z 1-2- z 3- z 1-2- z 2- z 3-2 z 1-2- z 3- z 2-2-sin- where 0- then - isA- -4B- 2 -3C- -6D- 5 -6

The correct option is A π/4Δ ABC is an equilateral ⇒ |z_2 z_1| = |z_3 z_1| = |z_2 z_2| = a and length of median =√(3)/2a = | z_2+z_3/2 z_1 | ∴ sin θ = √(a^ ...

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Byju's Answer

Standard XII

Mathematics

Definition of a Determinant

If z 1, z 2, ...

Question

If $z_{1}, z_{2}, z_{3}$ represent the vertices of an equilateral $Δ A B C$ and $\sqrt{\frac{| z_{2} - z_{1} |^{2} + | z_{3} - z_{1} |^{2}}{| z_{2} + z_{3} - 2 z_{1} |^{2} + | z_{3} - z_{2} |^{2}}} = s i n θ$ (where $0 < θ < π$ ) then $θ$ is

\frac{π}{4}

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\frac{2 π}{3}

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\frac{π}{6}

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\frac{5 π}{6}

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Solution

The correct option is A $\frac{π}{4}$
$Δ A B C$ is an equilateral $\Rightarrow | z_{2} - z_{1} | = | z_{3} - z_{1} | = | z_{2} - z_{2} | = a$ and length of median $= \frac{\sqrt{3}}{2} a = ∣ ∣ \frac{z_{2} + z_{3}}{2} - z_{1} ∣ ∣$
$∴ s i n θ = \sqrt{\frac{a^{2} + a^{2}}{3 a^{2} + a^{2}}} = \frac{1}{\sqrt{2}} \Rightarrow θ = \frac{π}{4}$

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Similar questions

Q. If

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represent the vertices of an equilateral triangle such that

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If z1,z2,z3 represent the vertices of an equilateral ΔABC and √|z2−z1|2+|z3−z1|2|z2+z3−2z1|2+|z3−z2|2=sin θ (where 0<θ<π) then θ is

The correct option is A π4ΔABC is an equilateral ⇒|z2−z1|=|z3−z1|=|z2−z2|=a and length of median =√32a=∣∣z2+z32−z1∣∣ ∴sin θ=√a2+a23a2+a2=1√2⇒θ=π4

If $z_{1}, z_{2}, z_{3}$ represent the vertices of an equilateral $Δ A B C$ and $\sqrt{\frac{| z_{2} - z_{1} |^{2} + | z_{3} - z_{1} |^{2}}{| z_{2} + z_{3} - 2 z_{1} |^{2} + | z_{3} - z_{2} |^{2}}} = s i n θ$ (where $0 < θ < π$ ) then $θ$ is