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Byju's Answer
Standard XII
Mathematics
Geometrical Representation of a Complex Number
If | z -2- i ...
Question
If
|
z
−
2
−
i
|
=
|
z
|
∣
∣
∣
sin
(
π
4
−
arg
(
z
)
)
∣
∣
∣
, where
z
is in first quadrant, then the locus of
z
is
A
a ray.
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B
a part of a circle.
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C
a part of a parabola.
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D
a part of an ellipse.
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Solution
The correct option is
C
a part of a parabola.
Given :
|
z
−
2
−
i
|
=
|
z
|
∣
∣
∣
sin
(
π
4
−
arg
(
z
)
)
∣
∣
∣
Let
z
=
x
+
i
y
and
arg
(
z
)
=
θ
⇒
tan
θ
=
y
x
, so
|
x
−
2
+
i
(
y
−
1
)
|
=
|
z
|
∣
∣
∣
sin
(
π
4
−
arg
(
z
)
)
∣
∣
∣
⇒
√
(
x
−
2
)
2
+
(
y
−
1
)
2
=
√
x
2
+
y
2
×
|
x
−
y
|
√
2
√
x
2
+
y
2
⇒
√
(
x
−
2
)
2
+
(
y
−
1
)
2
=
|
x
−
y
|
√
2
This represents a parabola, whose focus is
(
2
,
1
)
and equation of directrix is
y
=
x
Hence, the locus of
z
is a part of a parabola.
Suggest Corrections
2
Similar questions
Q.
If
|
z
−
2
−
i
|
=
|
z
|
∣
∣
∣
sin
(
π
4
−
arg
(
z
)
)
∣
∣
∣
, where
z
is in first quadrant, then the locus of
z
is
Q.
If the imaginary part of
2
z
+
1
i
z
+
1
is
−
2
,
then the locus of
z
is
Q.
The locus represented by the complex equation
|
z
−
2
−
i
|
=
|
z
|
sin
(
π
4
−
a
r
g
(
z
)
)
is the part of
Q.
If
|
z
−
2
−
i
|
=
|
z
|
|
sin
(
π
4
−
a
r
g
z
)
|
, then locus of
z
is
Q.
If the imaginary part of
z
−
4
z
−
2
i
is
0
, then the locus of
z
is
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Geometrical Representation of a Complex Number
Standard XII Mathematics
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