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The modulus of the complex number z such that | z + 3 – i | = 1 and arg(z) = π is equal to
- a)3
- b)2
- c)9
- d)4
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
The modulus of the complex number z such that | z + 3 – i | = 1 ...
Let z = x + iy
∴ | z + 3 – i | = | (x + 3) + i(y – 1) | = 1
∴ | z + 3 – i | = | (x + 3) + i(y – 1) | = 1
From equations (i) and (ii), we get x = –3, y = 0 ∴ z = –3
⇒ | z | = | –3 | = 3
⇒ | z | = | –3 | = 3
Most Upvoted Answer
The modulus of the complex number z such that | z + 3 – i | = 1 ...
Given Information:
| z + 3 – i | = 1
arg(z) = π
Solution:
- Conversion to Polar Form:
Let z = x + yi
| z + 3 – i | = 1
| x + yi + 3 – i | = 1
| x + 3 + (y – 1)i | = 1
|x + 3|² + (y – 1)² = 1 (Using modulus formula)
(x + 3)² + (y – 1)² = 1 (Squaring both sides)
- Substitute the given condition:
arg(z) = π
Argument of z is π which means z lies on the negative x-axis
- Substitute x = -3:
From the above equation, we know that x + 3 = 0
Therefore, x = -3
- Calculate Modulus:
| z | = | x + yi |
| z | = |-3 + yi|
| z | = √((-3)² + y²)
| z | = √(9 + y²)
- Substitute x = -3:
From the above equation, x = -3
| z | = √(9 + y²)
- Substitute in the given equation:
Substitute x = -3 in | z + 3 – i | = 1
| -3 + yi + 3 – i | = 1
| yi – i | = 1
| i(y – 1) | = 1
| y – 1 | = 1
y = 0 or y = 2
- Calculate Modulus:
Substitute y = 0, | z | = √(9 + 0) = 3
Substitute y = 2, | z | = √(9 + 4) = √13
Since z lies on the negative x-axis, the modulus of z is 3.
Therefore, the correct answer is option A.
| z + 3 – i | = 1
arg(z) = π
Solution:
- Conversion to Polar Form:
Let z = x + yi
| z + 3 – i | = 1
| x + yi + 3 – i | = 1
| x + 3 + (y – 1)i | = 1
|x + 3|² + (y – 1)² = 1 (Using modulus formula)
(x + 3)² + (y – 1)² = 1 (Squaring both sides)
- Substitute the given condition:
arg(z) = π
Argument of z is π which means z lies on the negative x-axis
- Substitute x = -3:
From the above equation, we know that x + 3 = 0
Therefore, x = -3
- Calculate Modulus:
| z | = | x + yi |
| z | = |-3 + yi|
| z | = √((-3)² + y²)
| z | = √(9 + y²)
- Substitute x = -3:
From the above equation, x = -3
| z | = √(9 + y²)
- Substitute in the given equation:
Substitute x = -3 in | z + 3 – i | = 1
| -3 + yi + 3 – i | = 1
| yi – i | = 1
| i(y – 1) | = 1
| y – 1 | = 1
y = 0 or y = 2
- Calculate Modulus:
Substitute y = 0, | z | = √(9 + 0) = 3
Substitute y = 2, | z | = √(9 + 4) = √13
Since z lies on the negative x-axis, the modulus of z is 3.
Therefore, the correct answer is option A.
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The modulus of the complex number z such that | z + 3 – i | = 1 and arg(z) = π is equal toa)3b)2c)9d)4Correct answer is option 'A'. Can you explain this answer?
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