Let - and - are complex numbers satisfying -1- i -1 and -2-3 i -6 such that 6-max-max- a - b - a - b - R -thenthe value of - b 2-2 a isA- 3B- 5C- 7D- 8

|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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Byju's Answer

Standard XII

Mathematics

Complex Numbers

Let α and β a...

Question

Let $α$ and $β$ are complex numbers satisfying $| α + 1 + i | = 1$ and $| β - 2 - 3 i | = 6$ such that $6 | α |_{m a x} - | β |_{m a x} = \sqrt{a} - \sqrt{b}; a, b \in R^{+}$ then the value of $\sqrt{b^{2} - 2 a}$ is

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Solution

The correct option is B $5$
$| α + 1 + i | = 1$
$\Rightarrow | α - (- 1 - i) | = 1$
$α$ represents circle with centre at $C_{1} (- 1, - 1)$ and Radius = $1$ unit.

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

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

Let $α$ and $β$ are complex numbers satisfying $| α + 1 + i | = 1$ and $| β - 2 - 3 i | = 6$ such that $6 | α |_{m a x} - | β |_{m a x} = \sqrt{a} - \sqrt{b}; a, b \in R^{+}$ then the value of $\sqrt{b^{2} - 2 a}$ is