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Mathematics Question on Complex Numbers and Quadratic Equations

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

A

y=1±23−xy = 1 \pm 2 \sqrt{3 - x}

B

y=2±3−xy = 2 \pm \sqrt{3 - x}

C

y=1±32−xy = 1 \pm 3 \sqrt{ 2 - x}

D

y=3±2−xy = 3 \pm \sqrt{2 - x}

Answer

y=1±23−xy = 1 \pm 2 \sqrt{3 - x}

Explanation

Solution

We have,
z=x+iyz = x +iy
and ∣z−(2+i)∣=∣Re(z)−4∣\left|z-\left(2+i\right)\right| = \left|Re\left(z\right)-4\right|
⇒∣x+iy−2−i∣=∣x−4∣\Rightarrow \left|x+iy -2 -i\right| = \left|x-4\right|
⇒∣(x−2)+(y−1)i∣=∣x−4∣\Rightarrow\left|\left(x-2\right)+\left(y-1\right)i\right|=\left|x-4\right|
⇒(x−2)2+(y−1)2=(x−4)2\Rightarrow \left(x-2\right)^{2} +\left(y-1\right)^{2} = \left(x-4\right)^{2}
⇒x2−4x+4+y2−2y+1=x2−8x+16\Rightarrow x^{2} -4x +4 + y^{2} -2y +1 = x^{2} -8x +16
⇒y2−2y+1=12−4x\Rightarrow y^{2} -2y +1 = 12 -4x
⇒(y−1)2=12−4x\Rightarrow \left(y-1\right)^{2} = 12 -4x
⇒y−1=±12−4x\Rightarrow y -1 = \pm\sqrt{12 -4x}
⇒y=1±23−x\Rightarrow y =1 \pm 2\sqrt{3-x}