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?
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?
- $y = 1 \pm 2 \sqrt{3 - x}$
- $y = 2 \pm \sqrt{3 - x}$
- $y = 1 \pm 3 \sqrt{ 2 - x}$
- $y = 3 \pm \sqrt{2 - x}$
The Correct Option is A
Solution and Explanation
$ z = x +iy $
and $\left|z-\left(2+i\right)\right| = \left|Re\left(z\right)-4\right| $
$ \Rightarrow \left|x+iy -2 -i\right| = \left|x-4\right| $
$ \Rightarrow\left|\left(x-2\right)+\left(y-1\right)i\right|=\left|x-4\right| $
$ \Rightarrow \left(x-2\right)^{2} +\left(y-1\right)^{2} = \left(x-4\right)^{2} $
$ \Rightarrow x^{2} -4x +4 + y^{2} -2y +1 = x^{2} -8x +16 $
$\Rightarrow y^{2} -2y +1 = 12 -4x$
$ \Rightarrow \left(y-1\right)^{2} = 12 -4x $
$ \Rightarrow y -1 = \pm\sqrt{12 -4x} $
$ \Rightarrow y =1 \pm 2\sqrt{3-x}$
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Concepts Used:
Complex Numbers and Quadratic Equations
Complex Number: Any number that is formed as a+ib is called a complex number. For example: 9+3i,7+8i are complex numbers. Here i = -1. With this we can say that i² = 1. So, for every equation which does not have a real solution we can use i = -1.
Quadratic equation: A polynomial that has two roots or is of the degree 2 is called a quadratic equation. The general form of a quadratic equation is y=ax²+bx+c. Here a≠0, b and c are the real numbers.
