Given a point Az in the Argand plane- Bz- and Cz- are points obtained on rotating Az about the Origin through an angle - anticlockwise and clockwise- respectively- without changing modulus of z- Identify the correct statements-

Z'=ze^i α, z^"=ze^ i α z'z^" = z^2 ⇒ z',z,z^" are in G.P. ( z'/z )^2 + ( z^"/z )^2=2 cos 2 α ⇒ (z')^2 + (z^")^2 = 2z^2 cos 2 α and z' + z^"=2z cos α ...

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Byju's Answer

Standard XII

Mathematics

Rotation Concept

Given a point...

Question

Given a point $A (z)$ in the Argand plane. $B (z^{'})$ and $C (z^{''})$ are points obtained on rotating $A (z)$ about the Origin through an angle $α$ anticlockwise and clockwise, respectively, without changing modulus of $z$ . Identify the correct statement(s).

z^{'}, z, z^{''}

form a G.P.

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z^{'}, z, z^{''}

form an A.P.

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(z^{'})^{2} + (z^{''})^{2} = 2 z^{2} c o s 2 α

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z^{'} + z^{''} = 2 z c o s 2 α

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Solution

The correct option is C $(z^{'})^{2} + (z^{''})^{2} = 2 z^{2} c o s 2 α$
$z^{'} = z e^{i α}, z^{''} = z e^{- i α}$

$z^{'} z^{''} = z^{2} \Rightarrow z^{'}, z, z^{''}$ are in G.P.

${(\frac{z^{'}}{z})}^{2} + {(\frac{z^{''}}{z})}^{2} = 2 cos 2 α$
$\Rightarrow (z^{'})^{2} + (z^{''})^{2} = 2 z^{2} cos 2 α$

and $z^{'} + z^{''} = 2 z cos α$

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Similar questions

Q. Given a point

A (z)

in the Argand plane.

B (z^{'})

and

C (z^{''})

are points obtained on rotating

A (z)

about the Origin through an angle

α

anticlockwise and clockwise, respectively, without changing modulus of

z

. Identify the correct statement(s).

Q. If the points

A (z), B (- z)

and

C (z + 1)

are vertices of an equilateral triangle then

Q. If

z_{1}

and

z_{2}

are two distinct points in an argand plane such that

a | z_{1} | = b | z_{2} |

(where

a, b \in R)

, then the point

(\frac{a z_{1}}{b z_{2}} + \frac{b z_{2}}{a z_{1}})

will always lie on the

Q. A complex number

z

is rotated in anticlockwise direction by an angle

α

and we get

z^{'}

, and if the same complex number

z

is rotated by an angle

α

in clockwise direction and we get

z^{''}

, then

Q. A complex number z is rotated in anticlockwise direction by an angle

α

and we get z' and if the same complex number z is rotated by an angle

α

in clockwise direction and we get z", then

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Given a point A(z) in the Argand plane. B(z′) and C(z′′) are points obtained on rotating A(z) about the Origin through an angle α anticlockwise and clockwise, respectively, without changing modulus of z. Identify the correct statement(s).

The correct option is C (z′)2+(z′′)2=2z2 cos 2αz′=zeiα, z′′=ze−iα z′z′′=z2⇒z′,z,z′′ are in G.P. (z′z)2+(z′′z)2=2 cos 2α ⇒(z′)2+(z′′)2=2z2 cos 2α and z′+z′′=2z cos α

Given a point $A (z)$ in the Argand plane. $B (z^{'})$ and $C (z^{''})$ are points obtained on rotating $A (z)$ about the Origin through an angle $α$ anticlockwise and clockwise, respectively, without changing modulus of $z$ . Identify the correct statement(s).