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Modulus of a Complex Number

The distance ...

Question

The distance between two complex numbers $z_{1} = x_{1} + i y_{2}$ and $z_{2} = x_{2} + i y_{2},$ is given by:

A

| (x_{1} - x_{2}) + i (y_{1} - y_{2}) |

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B

\sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}

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C

| z_{1} - z_{2} |

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D

\sqrt{(x_{1} - x_{2})^{2} - (y_{1} - y_{2})^{2}}

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Solution

The correct option is C $| z_{1} - z_{2} |$
$Given: z_{1} = x_{1} + i y_{2} and z_{2} = x_{2} + i y_{2}$

The distance between two complex numbers $z_{1}$ and $z_{2}$ is given by $| z_{1} - z_{2} | .$

$Here, | z_{1} - z_{2} | = | (x_{1} - x_{2}) + i (y_{1} - y_{2}) |$
$\Rightarrow | z_{1} - z_{2} | = \sqrt{(x_{1} - x_{2})^{2} + (y_{1} - y_{2})^{2}}$

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Q. Let z₁ and z₂ be two complex numbers such that |z₁ + z₂| = |z₁| + |z₂|, then arg (z₁) – arg (z₂) = ____________.

z_{1} = x_{1} + i y_{1}, z_{2} = x_{2} + i y_{2}

2 i ∣ ∣ ∣ \begin{matrix} x_{2} & y_{2} x_{1} & y_{1} \end{matrix} ∣ ∣ ∣

Q. For complex number

z_{1} = x_{1} + i y_{1}

z_{2} = x_{2} + i y_{2}

z_{1} \cap z_{2}

x_{1} \leq x_{2}

y_{1} \leq y_{2}

. Then for all complex numbers z with

1 \cap z

((1 - z) / (1 + z)) \cap 0

. If this is true enter 1, else enter 0.

x_{1}, x_{2}, y_{1}, y_{2} ϵ R

x_{1}

x_{2}, y_{1} = y_{2}

z_{1} = x_{1} + i y_{1}, z_{2} = x_{2} + i y_{2}

z_{3} = \frac{1}{2} (z_{1} + z_{2})

z_{1}, z_{2}, z_{3}

For complex numbers

z_{1} = x_{1} + i y_{1}

z_{2} = x_{2} + i y_{2}

z_{1} \cap z_{2}

x_{1} \leq x_{2}

y_{1} \leq y_{2}

. Then, for all complex numbers

z

1 \cap z

\frac{1 - z}{1 + z} \cap 0

x^{2} + y^{2} \geq 0

y \geq 0

.If true enter 1 else 0.

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