Q 10 If Z = 2 + i and K= 2 - 1, then IZ + Klis
Ops: A.2
B.6
C.8
D. 4
Answers
Answer:
If z
1
and z
2
are two complex numbers such that
∣
∣
∣
∣
∣
z
1
+z
2
z
1
−z
2
∣
∣
∣
∣
∣
=1,
z
2
iz
1
=k, where k is a real number. Find the angle between the lines from the origin to the points z
1
+z
2
and z
1
−z
2
in terms of k.
ANSWER
(i) Given
∣
∣
∣
∣
∣
z
1
+z
2
z
1
−z
2
∣
∣
∣
∣
∣
=1
⇒
∣
∣
∣
∣
∣
∣
z
2
z
1
+1
z
2
z
1
−1
∣
∣
∣
∣
∣
∣
=1
⇒
∣
∣
∣
∣
∣
z
2
z
1
−1
∣
∣
∣
∣
∣
=
∣
∣
∣
∣
∣
z
2
z
1
+1
∣
∣
∣
∣
∣
squaring both sides
∣
∣
∣
∣
∣
z
2
z
1
∣
∣
∣
∣
∣
2
+1−2Re(
z
2
z
1
)=
∣
∣
∣
∣
∣
z
2
z
1
∣
∣
∣
∣
∣
2
+1+2Re(
z
2
z
1
)
⇒4Re(
z
2
z
1
)0⇒
z
2
z
1
is purely imaginary number
z
2
z
1
can be written as i
z
2
z
1
=k where k is real number.
(ii) Let θ is the angle between z
1
−z
2
and z
1
−z
2
, then
θ=Arg(
z
1
−z
2
z
1
+z
2
)
=Arg
⎝
⎜
⎜
⎛
z
2
z
1
−1
z
2
z
1
+1
⎠
⎟
⎟
⎞
=Arg(
−ik−1
−ik+1
)
=Arg(
1+ik
−1+ik
)
=Arg(
k
2
+1
k
2
−1+2ik
)
θ=tan
−1
(
k
2
−1
2k
)
Step-by-step explanation:
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