Which one of the following is not true for complex number $z_{1}$ and $z_{2}$ ?

\frac{z_{1}}{z_{2}} = \frac{z_{1} {¯ ¯ ¯ z}_{1}}{| z_{2} |^{2}}

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| z_{1} + z_{2} | \leq | z_{1} | + | z_{2} |

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| z_{1} - z_{2} | \leq | z_{1} | - | z_{2} |

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| z_{1} + z_{2} |^{2} + | z_{1} - z_{2} |^{2} = 2 | z_{1} |^{2} + 2 | z_{2} |^{2}

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Solution

The correct option is C $| z_{1} - z_{2} | \leq | z_{1} | - | z_{2} |$
$\frac{z_{1} {¯ ¯ ¯ z}_{1}}{| z_{2} |^{2}} = \frac{z_{1} {¯ ¯ ¯ z}_{2}}{z_{2} {¯ ¯ ¯ z}_{2}} = \frac{z_{1}}{z_{2}};$ (given) i.e., (a) is correct
let $| Z_{1} | = O A = B C$ & $| Z_{2} | = O B = A C$
Now in the following figure,

then, $| z_{1} + z_{2} | = O C$ & $| z_{1} - z_{2} | = B A$
We know that in any triangle sum of any two sides the third side
i.e., $| z_{1} | + | z_{2} | \geq | z_{1} + z_{2} |$
Also, Difference of any two sides $\leq$ the third side
i.e., $| | z_{1} | - | z_{2} | | \leq | z_{1} - z_{2} |$ .
Hence, (c) is not true. Not that option (d) is true

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Similar questions

z_{1} : (3, 4) & z_{2} : (4, 5)

z_{1} + z_{2}

Which of the following is correct for any two complex number $z_{1}$ and $z_{2}$ ?

|z_{1} z_{2}| = |z_{1}| |z_{2}|

\arg (z_{1} z_{2}) = \arg (z_{1}) \arg (z_{2})

|z_{1} + z_{2}| = |z_{1}| + |z_{2}|

|z_{1} + z_{2}| \geq |z_{1}| + |z_{2}|

z_{1}

z_{2}

arg (z_{1}) - arg (z_{2}) = 0

| z_{1} - z_{2} |

z

| z_{1} + z_{2} |^{2} + | z_{1} - z_{2} |^{2} = 2 (| z_{1} |^{2} + | z_{2} |^{2})

z_{1}

z_{2} .

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