Let - and - be complex numbers satisfying - - 1 - i- - 1 and - - 2 - 3i- - 6 such that 6-max - -max - -a - -b- where a - b - Then the value of -b2 - 2a is

|α + 1 + i| = 1 ⇒ |α nbsp;( 1 i)| = 1 α represents circle with centre at C_1 nbsp;( 1, 1) and radius = 1 unit. |α| = |α 0| = distance from point o ...

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Standard XI

Mathematics

Properties of Modulus

Let α and β b...

Question

Let $α$ and $β$ be complex numbers satisfying $| α + 1 + i | = 1$ and $| β - 2 - 3 i | = 6$ such that $6 | α |_{max} - | β |_{max} = \sqrt{a} - \sqrt{b},$ where $a, b \in R^{+} .$ Then the value of $\sqrt{b^{2} - 2 a}$ is

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Solution

$| α + 1 + i | = 1$
$\Rightarrow | α - (- 1 - i) | = 1$
$α$ represents circle with centre at $C_{1} (- 1, - 1)$ and radius = $1$ unit.

$| α | = | α - 0 | =$ distance from point on circle to origin.
$| α |_{max} = O C_{1} + r$ $= \sqrt{2} + 1$
Similarly, for $| β - (2 + 3 i) | = 6$
$| β |_{max} = \sqrt{13} + 6$
So, $6 | α |_{max} - | β |_{max} = 6 \sqrt{2} - \sqrt{13} = \sqrt{72} - \sqrt{13}$
$∴ \sqrt{b^{2} - 2 a} = \sqrt{169 - 144} = 5$

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Properties of Modulus

Standard XI Mathematics

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Let α and β be complex numbers satisfying |α+1+i|=1 and |β−2−3i|=6 such that 6|α|max−|β|max=√a−√b, where a,b∈R+. Then the value of √b2−2a is

Let $α$ and $β$ be complex numbers satisfying $| α + 1 + i | = 1$ and $| β - 2 - 3 i | = 6$ such that $6 | α |_{max} - | β |_{max} = \sqrt{a} - \sqrt{b},$ where $a, b \in R^{+} .$ Then the value of $\sqrt{b^{2} - 2 a}$ is