Complex NumbersHard
Question
If z1 = a + ib and z2 = c + id are complex numbers such that |z1|= |z2| = 1 and Re (z1
2) = 0,
then the pair of complex numbers w1 = a + ic and w2 = b + id satifes
2) = 0, then the pair of complex numbers w1 = a + ic and w2 = b + id satifes
Options
A.|W1| = 1
B.|W2| = 1
C.Re (z12) = 0
2) = 0D.None of these
Solution
Since, z1 = a + ib and z2 = c + id
⇒ |z1|2 a2 + b2 = 1 and |z2|2 = c2 + d2 = 1 ......(i)
Also, Re (z12) = 0 ⇒ ac + bd =0
⇒
(say).....(ii)
From Eqs. (i) and (ii), b2λ2 + b2 = c2 + λ2c2
⇒ b2 = c2 and a2 = d2
Also, given w1 = a + ic and w2 = b + id
Now, |w1|
|w2|
and Re (w1
2)= ad + cb(bλ) b + c(-λc)
(w12)= ad + cb(bλ )b + c(-λc) [from Eq. (i)]
Therefore, (a), (b), (c) are the correct answers.
⇒ |z1|2 a2 + b2 = 1 and |z2|2 = c2 + d2 = 1 ......(i)
Also, Re (z1
2) = 0 ⇒ ac + bd =0 ⇒
(say).....(ii) From Eqs. (i) and (ii), b2λ2 + b2 = c2 + λ2c2
⇒ b2 = c2 and a2 = d2
Also, given w1 = a + ic and w2 = b + id
Now, |w1|
|w2|
and Re (w1
2)= ad + cb(bλ) b + c(-λc) (w1
2)= ad + cb(bλ )b + c(-λc) [from Eq. (i)]Therefore, (a), (b), (c) are the correct answers.
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