Let $z = x + i y$ be a complex number satisfying equation $| z - (2 + i) | = | R e (z) - 4 |$
Which of the following options describes the above equation ?

y = 1 \pm 2 \sqrt{3 - x}

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y = 2 \pm \sqrt{3 - x}

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y = 1 \pm 3 \sqrt{2 - x}

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y = 3 \pm \sqrt{2 - x}

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Solution

The correct option is A $y = 1 \pm 2 \sqrt{3 - x}$
$z = x + i y \Rightarrow | z - (2 + i) | = | R e (z) - 4 | \Rightarrow | x - 2 + i (y - 1) | = | x - 4 | \Rightarrow (x - 2)^{2} + (y - 1)^{2} = (x - 4)^{2} \Rightarrow (y - 1)^{2} = (x - 4)^{2} - (x - 2)^{2}$
$\Rightarrow (y - 1)^{2} = - 2 (2 x - 6) \Rightarrow y = 1 \pm 2 \sqrt{3 - x}$

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Let z=x+iy be a complex number satisfying equation |z−(2+i)|=|Re(z)−4| Which of the following options describes the above equation ?

The correct option is A y=1±2√3−xz=x+iy⇒|z−(2+i)|=|Re(z)−4|⇒|x−2+i(y−1)|=|x−4|⇒(x−2)2+(y−1)2=(x−4)2⇒(y−1)2=(x−4)2−(x−2)2 ⇒(y−1)2=−2(2x−6)⇒y=1±2√3−x

Let $z = x + i y$ be a complex number satisfying equation $| z - (2 + i) | = | R e (z) - 4 |$
Which of the following options describes the above equation ?