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The modulus of the complex number z such that |z+3-i|= 1 and arg (z)= π is equal to?
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The modulus of the complex number z such that |z+3-i|= 1 and arg (z)= ...
Modulus of a Complex Number with Given Conditions
To find the modulus of the complex number z with the given conditions, we will use the following steps:
Step 1: Write the Complex Number in Polar Form
We know that the modulus of a complex number is given by |z| = √(x² + y²) and the argument of a complex number is given by arg(z) = tan⁻¹(y/x). Therefore, we can write the complex number z in polar form as:
z = |z| (cosθ + i sinθ)
where |z| is the modulus of z and θ is the argument of z.
Step 2: Use the Given Conditions to Find |z| and θ
We are given that |z 3-i| = 1, which means that the distance between z and the point (3, -1) in the complex plane is 1. Therefore, we can write:
|z - (3 - i)| = 1
This represents a circle with center (3, -1) and radius 1 in the complex plane. We are also given that the argument of z is π, which means that z lies on the negative real axis.
Therefore, we can write z in polar form as:
z = |z| (cosπ + i sinπ) = -|z|
This means that the real part of z is -|z| and the imaginary part is 0.
Step 3: Find |z|
Substituting z in the equation |z - (3 - i)| = 1, we get:
|-|z| - (3 - i)| = 1
|-|z| + 3 - i| = 1
This represents a circle with center (-3, i) and radius 1 in the complex plane. We know that z lies on the negative real axis, which means that the distance between z and (-3, i) is |z| + 3.
Therefore, we can write:
|z| + 3 = 1
|z| = -2
Since the modulus of a complex number is always a positive number, this is not a valid solution. Therefore, there is no complex number z that satisfies the given conditions.
Final Answer:
There is no complex number z that satisfies the given conditions.
Community Answer
The modulus of the complex number z such that |z+3-i|= 1 and arg (z)= ...
Locus of Z is a circle of radius 1 unit and centered at (-3,1). As argument of Z is π it lies on -ve x-axis. Therefore Coordinate of Z is (-3,0) in imaginary and real axis. Therefore |Z| =3 units.....
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The modulus of the complex number z such that |z+3-i|= 1 and arg (z)= π is equal to?
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The modulus of the complex number z such that |z+3-i|= 1 and arg (z)= π is equal to? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The modulus of the complex number z such that |z+3-i|= 1 and arg (z)= π is equal to? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The modulus of the complex number z such that |z+3-i|= 1 and arg (z)= π is equal to?.
The modulus of the complex number z such that |z+3-i|= 1 and arg (z)= π is equal to? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The modulus of the complex number z such that |z+3-i|= 1 and arg (z)= π is equal to? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The modulus of the complex number z such that |z+3-i|= 1 and arg (z)= π is equal to?.
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