SATHEE: Complex Numbers 2 Question 24

Complex Numbers 2 Question 24

25. If $z_{1} = a + i b$ and $z_{2} = c + i d$ are complex numbers such that $| z_{1} | = | z_{2} | = 1$ and $Re (z_{1} {\bar{z}}_{2}) = 0$ , then the pair of complex numbers $w_{1} = a + i c$ and $w_{2} = b + i d$ satisfies

(a) $| w_{1} | = 1$

(b) $| w_{2} | = 1$

(d) None of these

$(1985, 2 M)$

Passage Based Problems

Read the following passages and answer the questions that follow.

Passage I

Let $A, B, C$ be three sets of complex number as defined below

$\begin{aligned} A = z : lm (z) \geq 1 \\ B = z : | z - 2 - i | = 3 \\ C = z : Re ((1 - i) z) = \sqrt{2} \end{aligned}$

(2008, 12M)

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Answer:

Correct Answer: 25. (b)

Solution:

Since, $z_{1} = a + i b$ and $z_{2} = c + i d$

$\Rightarrow {| z_{1} |}^{2} = a^{2} + b^{2} = 1$ and ${| z_{2} |}^{2} = c^{2} + d^{2} = 1$

$[∵ | z_{1} | = | z_{2} | = 1]$

Also, $Re (z_{1} {\bar{z}}_{2}) = 0 \Rightarrow a c + b d = 0$

$\Rightarrow \frac{a}{b} = - \frac{d}{c} = λ$

From Eqs. (i) and (ii), $b^{2} λ^{2} + b^{2} = c^{2} + λ^{2} c^{2}$

$\Rightarrow b^{2} = c^{2}$ and $a^{2} = d^{2}$

Also, given $w_{1} = a + i c$ and $w_{2} = b + i d$

Now, $| w_{1} | = \sqrt{a^{2} + c^{2}} = \sqrt{a^{2} + b^{2}} = 1$

$| w_{2} | = \sqrt{b^{2} + d^{2}} = \sqrt{a^{2} + b^{2}} = 1$

and $Re (w_{1} \overset{―}{w_{2}}) = a b + c d = (b λ) b + c (- λ c)$ [from Eq. (i)]

$= λ (b^{2} - c^{2}) = 0$

पिछला

अगला

Complex Numbers 2 Question 24

25. If $z_{1} = a + i b$ and $z_{2} = c + i d$ are complex numbers such that $| z_{1} | = | z_{2} | = 1$ and $Re (z_{1} {\bar{z}}_{2}) = 0$ , then the pair of complex numbers $w_{1} = a + i c$ and $w_{2} = b + i d$ satisfies

Passage Based Problems

Passage I

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Complex Numbers 2 Question 24

25. If z1=a+ib and z2=c+id are complex numbers such that |z1|=|z2|=1 and Re(z1z¯2)=0, then the pair of complex numbers w1=a+ic and w2=b+id satisfies

Passage Based Problems

Passage I

Answer:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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25. If $z_{1} = a + i b$ and $z_{2} = c + i d$ are complex numbers such that $| z_{1} | = | z_{2} | = 1$ and $Re (z_{1} {\bar{z}}_{2}) = 0$ , then the pair of complex numbers $w_{1} = a + i c$ and $w_{2} = b + i d$ satisfies