Suppose z 1- z 2- z 3 are the vertices of an equilateral triangle inscribed in the circle - z -2- If z 1-1- i -3- then values of z 3 and z 2 are respectively A- -2-3B- 2-1-i -3- 1-i -3- 2D- None of these

One of the number must be a conjugate of z_1=1+i√(3) i.e. z_2=1 √(3) or z_3=z_1e^i2x/3 and z_2=z_1e^ i2x3 z_3=(1+i√(3))[cos(2π/3 )+i sin2π/3]= 2 Aliter : O ...

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Byju's Answer

Standard XII

Mathematics

Circle Inscribing a Triangle Fromed by 3 Given Lines

Suppose z 1, ...

Question

Suppose $z_{1}, z_{2}, z_{3}$ are the vertices of an equilateral triangle inscribed in the circle $| z | = 2$ . If $z_{1} = 1 + i \sqrt{3}$ . then values of $z_{3} a n d z_{2}$ are respectively

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Solution

The correct option is A
One of the number must be a conjugate of
$z_{1} = 1 + i \sqrt{3} i . e . z_{2} = 1 - \sqrt{3} o r z_{3} = z_{1} e^{i 2 x / 3} a n d z_{2} = z_{1} e^{- i 2 x 3} z_{3} = (1 + i \sqrt{3}) [c o s (\frac{2 π}{3}) + i s i n \frac{2 π}{3}] = - 2$
Aliter : Obviously $| z | = 2$ is a circle with centre o (0, 0) and radius 2. Therefore, OA=OB=OC and this is satisfied by (a) because two vertices of any triangle cannot be same.

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Suppose z1,z2,z3 are the vertices of an equilateral triangle inscribed in the circle |z|=2. If z1=1+i√3. then values of z3 and z2 are respectively

Suppose $z_{1}, z_{2}, z_{3}$ are the vertices of an equilateral triangle inscribed in the circle $| z | = 2$ . If $z_{1} = 1 + i \sqrt{3}$ . then values of $z_{3} a n d z_{2}$ are respectively