If for z = i , | z + 2 | = z + 4 ( 1 + i ) , then and are the roots of the equation [2023]

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Solution

Q.
$If for z = α + i β, | z + 2 | = z + 4 (1 + i), then α + β and α β are the roots of the equation$ [2023]

1 $x^{2} + 7 x + 12 = 0$

2 $x^{2} + x - 12 = 0$

3 $x^{2} + 3 x - 4 = 0$

4 $x^{2} + 2 x - 3 = 0$

Ans.
(1)

We have, $| z + 2 |$ $= z + 4 (1 + i)$

$\Rightarrow$ $| α + i β + 2 |$ $= α + i β + 4 + 4 i$

$\Rightarrow$ $\sqrt{{(α + 2)}^{2} + β^{2}} = (α + 4) + i (β + 4)$

Comparing real and imaginary parts, we get

$β + 4 = 0 \Rightarrow β = - 4$

Also, ${(α + 2)}^{2} + 4^{2} = {(α + 4)}^{2}$

$\Rightarrow α^{2} + 4 + 4 α + 16 = α^{2} + 16 + 8 α \Rightarrow 4 α = 4 \Rightarrow α = 1$

Now, $α + β = - 3, α β = - 4$

Equation having roots - 3 and - 4 is $(x + 3) (x + 4) = 0$

i.e. $x^{2} + 7 x + 12 = 0$

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