The minimum value of $| Z - 1 + 2 i | + | 4 i - 3 - Z |$ is

\sqrt{5}

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5

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2 \sqrt{13}

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\sqrt{15}

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Solution

The correct option is A $\sqrt{5}$
$| Z - 1 + 2 i | + | 4 i - 2 - Z | \geq {| Z | - | (1 - 2 i) |} + {| (4 i - 2) | - | Z |}$ [As $| a - b | \geq | a | - | b |$ ]
or, $| Z - 1 + 2 i | + | 4 i - 2 - Z | \geq - | (1 - 2 i) | + | (4 i - 2) |$
or, $| Z - 1 + 2 i | + | 4 i - 2 - Z | \geq - \sqrt{5} + \sqrt{20}$
or, $| Z - 1 + 2 i | + | 4 i - 2 - Z | \geq - \sqrt{5} + 2 \sqrt{5} = \sqrt{5}$ .
So the minimum value is $\sqrt{5}$ .

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Similar questions

The minimum value of $| z - 1 + 2 i | + | 4 i - 3 - z |$ is

z_{1} = 5 + 12 i

| z_{2} | = 4

| z - 3 - 4 i |^{2} + | z + 2 - 7 i |^{2} + | z - 5 + 2 i |^{2}

z = x + i y

x

z_{1}, z_{2}

| z_{1} | = 12; | z_{2} - 3 - 4 i | = 5

| z_{1} - z_{2} |

z_{1} = 5 + 5 i, z_{2} = - 4 + 4 i, z_{3} = - 3 - 3 i

z_{4} = 2 - 2 i,

i = \sqrt{- 1}

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