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Geometrical Representation of a Complex Number

For two compl...

Question

For two complex numbers $z_{1}$ and $z_{2}$ : $(a z_{1} + b_{1}) (c z_{2} + d_{2}) = (c z_{1} + d_{1}) (a z_{2} + b_{2})$ if $(a, b, c, d \in R)$

A

\frac{a}{b} = \frac{c}{d}

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B

\frac{a}{d} = \frac{b}{c}

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C

| z_{1} | = | z_{2} |

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D

a r g (z_{1}) = a r g (z_{2})

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Solution

The correct options are
A $\frac{a}{b} = \frac{c}{d}$
C $| z_{1} | = | z_{2} |$

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Similar questions

z_{1}, z_{2}

are complex numbers then the correct match List- I to List II is:

List - I	List - II
A) arg z_1z_2	1) $a r g z_{1} - a r g z_{2}$
B) $a r g z_{1}_{2}$	2) $a r g z_{1} - a r g z_{2} = \frac{π}{2}$
C) $\| z_{1} + z_{2} \| = \| z_{1} - z_{2} \|$	3) $a r g z_{1} = a r g z_{2}$
D) $\| z_{1} + z_{2} \|^{2}$	4) $a r g z_{1} + a r g z_{2}$
E) $\| z_{1} + z_{2} \| = \| z_{1} \| + \| z_{2} \|$	5) $r_{1}^{2} + r_{2}^{2} + 2 r_{1} r_{2} c o s (θ_{1} - θ_{2})$

Q. For two complex numbers

z_{1}

z_{2} : (a z_{1} + b {¯ ¯ ¯ z}_{1}) (c z_{2} + d {¯ ¯ ¯ z}_{2}) = (c z_{1} + d {¯ ¯ ¯ z}_{1}) (a z_{2} + b {¯ ¯ ¯ z}_{2})

(a, b, c, d ϵ R)

Q. For two complex numbers

z_{1}

z_{2}; (a z_{1} + b_{1}) (c z_{2} + d_{2}) = (c z_{1} + d_{1}) (a z_{2} + b_{2})

b \neq 0, d \neq 0

Q. For any two complex number

z_{1}, z_{2}

and any two real numbers

a

b

| a z_{1} - b z_{2} |^{2} + | b z_{1} + a z_{2} |^{2} =

z_{1}

z_{2}

are any two complex numbers, then

∣ ∣ ∣ z_{1} + \sqrt{z_{1}^{2} - z_{2}^{2}} ∣ ∣ ∣ + ∣ ∣ ∣ z_{1} - \sqrt{z_{1}^{2} - z_{2}^{2}} ∣ ∣ ∣

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