If z 1 is rotated through an angle of 120- anti-clockwise about z 0 to reach z 2- then z 2- z 0- z 1- z 0 equal toA- -1-i -3-2B- -1-i -3-2C- 1-i -3-2D- 1-i -3-2

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Standard XII

Mathematics

Four Common Forms of Parabola Equation

If z 1 is rot...

Question

If $z_{1}$ is rotated through an angle of $120^{\circ}$ anti-clockwise about $z_{0}$ to reach $z_{2}$ , then $\frac{z_{2} - z_{0}}{z_{1} - z_{0}}$ equal to

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Solution

The correct option is A

$\frac{z_{1} - z_{0}}{z_{2} - z_{0}}$ = $∣ ∣ \frac{z_{1} - z_{0}}{z_{2} - z_{0}} ∣ ∣$ $e^{i 2 \frac{π}{3}}$

| $z_{1}$ - $z_{0}$ | = | $z_{2}$ - $z_{0}$ |

$\Rightarrow$ $\frac{z_{1} - z_{0}}{z_{2} - z_{0}}$ = $e^{i 2 \frac{π}{3}}$

= $\frac{- 1 + i \sqrt{3}}{2}$

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Similar questions

Let $z_{1}, z_{2}, z_{3}$ are the vertices of an equilateral triangle inscribed in the circle $| z |$ = 2. If $z_{1}, z_{2}, z_{3}$ are in the clockwise sense and $z_{1}$ = $1 + i \sqrt{3}$ , then:

Q. If

| z_{1} - z_{0} | = | z_{2} - z_{0} | = a

and

a m p \frac{z_{2} - z_{0}}{z_{0} - z_{1}} = \frac{π}{2}

, then

z_{0}

is equal to

Q. Let

A (z_{1})

and

B (z_{2})

be two points lying on the curve

z - 3 - 4 i = \frac{25}{¯ ¯ ¯ z - 3 + 4 i}

where

| z_{1} |

is maximum. Now,

A (z_{1})

is rotated about the origin in anti-clockwise direction through

90^{\circ}

reaching at

P (z_{0}) .

A, B

and

P

are collinear, then the value of

| z_{0} - z_{1} | | z_{0} - z_{2} |

Q. I. If

\frac{z_{1}}{z_{2}} + \frac{z_{2}}{z_{1}} = 1

then origin

z_{1}, z_{2}

form equilateral triangle.
II.

z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 3 z_{0}^{2}

then

z_{0}

is circumcentre of equilateral triangle with the vertices

z_{1}, z_{2}, z_{3}

Q. Assertion :

Let

z_{1}, z_{2}, z_{3}

satisfy

∣ ∣ ∣ \frac{z + 2}{z - 1} ∣ ∣ ∣ = 2

and

z_{0} = 2

. Consider least positive arguments wherever required

2 arg (\frac{z_{1} - z_{3}}{z_{2} - z_{3}}) = arg (\frac{z_{1} - z_{0}}{z_{2} - z_{0}})

. Reason:

z_{1}, z_{2}, z_{3}

satisfy

| z - z_{0} | = 2

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If z1 is rotated through an angle of 120∘ anti-clockwise about z0 to reach z2 , then z2−z0z1−z0 equal to

The correct option is A z1−z0z2−z0 = ∣∣z1−z0z2−z0∣∣ ei2π3 |z1 - z0| = |z2 - z0| ⇒ z1−z0z2−z0 = ei2π3 = −1+i√32

If $z_{1}$ is rotated through an angle of $120^{\circ}$ anti-clockwise about $z_{0}$ to reach $z_{2}$ , then $\frac{z_{2} - z_{0}}{z_{1} - z_{0}}$ equal to

The correct option is A

$\frac{z_{1} - z_{0}}{z_{2} - z_{0}}$ = $∣ ∣ \frac{z_{1} - z_{0}}{z_{2} - z_{0}} ∣ ∣$ $e^{i 2 \frac{π}{3}}$

| $z_{1}$ - $z_{0}$ | = | $z_{2}$ - $z_{0}$ |

$\Rightarrow$ $\frac{z_{1} - z_{0}}{z_{2} - z_{0}}$ = $e^{i 2 \frac{π}{3}}$

= $\frac{- 1 + i \sqrt{3}}{2}$