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Byju's Answer
Standard XII
Mathematics
Cos(A+B)Cos(A-B)
The complex n...
Question
The complex number z having least positive argument which satisfies the condition
|
z
−
25
i
|
≤
15
is:
A
25
i
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B
12
+
5
i
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C
16
+
12
i
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D
12
+
16
i
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Solution
The correct option is
D
12
+
16
i
Solution:
|
z
−
25
i
|
≤
15
Let
z
=
r
(
c
o
s
θ
+
i
s
i
n
θ
)
θ
must be minimum
|
r
c
o
s
θ
+
i
(
r
s
i
n
θ
−
25
)
|
≤
15
|
√
r
2
c
o
s
2
θ
+
r
2
s
i
n
2
θ
+
625
−
50
r
s
i
n
θ
|
≤
15
square both side
r
2
(
c
o
s
2
θ
+
s
i
n
2
θ
)
+
625
−
50
r
s
i
n
θ
≤
225
r
2
50
r
s
i
n
θ
≤
−
400
f
(
r
)
=
400
+
r
2
50
r
≤
s
i
n
θ
Find maximum value of
f
(
r
)
=
400
+
r
2
50
r
f
′
(
r
)
=
100
r
2
−
50
(
400
+
r
2
)
2500
r
2
=
0
50
r
2
−
50
×
400
=
0
r
=
20
f
(
r
=
20
)
=
800
1000
≤
s
i
n
θ
4
5
≤
s
i
n
θ
Least value of
s
i
n
θ
is
4
/
5
t
a
n
θ
=
4
/
3
c
o
s
θ
=
3
/
5
z
=
20
(
3
/
5
+
4
/
5
i
)
z
=
12
+
16
i
D is correct.
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