If z is any complex number satisfying - z - 3 - 2i - - 2- then the minimum value of - 2z - 6 - 5i - is

From the given condition, | z – 3 – 2i | ≤ nbsp;2 ⇒ nbsp;| 2z – 6 – 4i | ≤ nbsp;4 ⇒ nbsp;4 ≥ nbsp;| (2z – 6 + 5i) – 9i | ≥ nbsp;|| 9i | – | 2z – 6 ...

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Standard XI

Mathematics

Modulus of a Complex Number

If z is any c...

Question

If $z$ is any complex number satisfying $| z - 3 - 2 i | \leq 2$ , then the minimum value of $| 2 z - 6 + 5 i |$ is

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Solution

From the given condition,

$| z - 3 - 2 i | \leq 2$
$\Rightarrow | 2 z - 6 - 4 i | \leq 4$
$\Rightarrow 4 \geq | (2 z - 6 + 5 i) - 9 i |$
$\geq | | 9 i | - | 2 z - 6 + 5 i | |$
$\Rightarrow 4  9 - | 2 z - 6 + 5 i |$
$\Rightarrow | 2 z - 6 + 5 i | \geq 5$

Minimum value of $| 2 z - 6 + 5 i | i s = 5$

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If z is any complex number satisfying |z–3–2i|≤2, then the minimum value of |2z–6+5i| is

From the given condition, |z–3–2i|≤2 ⇒|2z–6–4i|≤4 ⇒4≥|(2z–6+5i)–9i| ≥||9i|–|2z–6+5i|| ⇒49–|2z–6+5i| ⇒|2z–6+5i|≥5 Minimum value of |2z–6+5i|is=5

If $z$ is any complex number satisfying $| z - 3 - 2 i | \leq 2$ , then the minimum value of $| 2 z - 6 + 5 i |$ is