If - Z - i - - 2 and Z 1-5-3 i- then the maximum value of - iZ - Z 1- isA- 7 - 9 - 7-13D- 7-12

We have, |Z i|≤2 Let z = x+iy, then nbsp; ⇒ x^2 +(y 1)^2 ≤ 4 So, nbsp;Z lies on or inside the circle, whose center is (0,1) and radius is 2 nbsp;units No ...

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Standard XII

Mathematics

Equation of Circle in Complex Form

If | Z - i | ...

Question

If $| Z - i | \leq 2$ and $Z_{1} = 5 + 3 i$ , then the maximum value of $| i Z + Z_{1} |$ is

7 + \sqrt{13}

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7 + \sqrt{12}

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7

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9

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Solution

The correct option is C $7$
We have,
$| Z - i | \leq 2$
Let $z = x + i y$ , then
$\Rightarrow x^{2} + (y - 1)^{2} \leq 4$
So, $Z$ lies on or inside the circle, whose center is $(0, 1)$ and radius is $2 units$

Now,
$| i Z + Z_{1} | = | i Z - i^{2} Z_{1} | = | i (Z - i (5 + 3 i)) | = | i | | Z - (- 3 + 5 i) | = | Z - (- 3 + 5 i) | (∵ | i | = 1)$

Point $(- 3, 5)$ lies outside the circle, so the maximum value of $| i Z + Z_{1} |$
$= \sqrt{(0 + 3)^{2} + (1 - 5)^{2}} + 2 = 7$

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