Conjugates of Complex Numbers MCQ - Practice Questions & Answers

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Let $a \neq b$ be two-zero real numbers. Then the number of elements in the set $X=\left\{z \in \mathbb{C}: \operatorname{Re}\left(a z^2+b z\right)=\mathrm{a}\right.$ and $\mathrm{\left.\operatorname{Re}\left(b z^2+\mathrm{az}\right)=\mathrm{b}\right\}}$ is equal to :

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Conjugate of complex numbers and their properties

The conjugate of a complex number z = a + ib (a, b are real numbers) is a − ib. It is denoted as $\bar{z}$ .

i.e. if z = a + ib, then its conjugate is $\bar{z}$ = a - ib.

Conjugate of complex numbers is obtained by changing the sign of the imaginary part of the complex number. The real part of the number is left unchanged.

Note:

When a complex number is added to its complex conjugate, the result is a real number. i.e. z = a + ib, $\bar{z}$ = a - ib

Then the sum, z + $\bar{z}$ = a + ib + a - ib = 2a (which is real)

When a complex number is multiplied by its complex conjugate, the result is a real number i.e. z = a + ib, $\bar{z}$ = a - ib

Then the product, z・ $\bar{z}$ = (a + ib)・(a - ib) = a² - (ib)²

= a² + b²(which is real)

Geometrically complex conjugate of a complex number is its mirror image with respect to the real axis (x-axis).

For example

z = 2 + 2i and $\bar{z}$ = 2 - 2i

Properties of the conjugate complex numbers:

z, z₁, z_2,and z₃be the complex numbers

$\\\mathrm{1.\;\overline{\left(\overline{z}\right)}=z}\\\mathrm{2.\;z+\bar z=2.Re(z)}\\\mathrm{3.\;z-\bar z=2i.Im(z)}\\\mathrm{4.\;z+\bar z=0\;\Rightarrow \;z=-\bar z\;\Rightarrow z\;is\;purely \;imaginary}\\\mathrm{5.\;z-\bar z=0\Rightarrow z=\bar z\;\Rightarrow z\;is\;purely\;real}$

$\\\mathrm{6.\;\overline{z_1\pm z_2}\:=\:\overline{z_1}\pm\overline{z_2}}\\\\\mathrm{\;\;\;In\,\,general,\;\;\overline{z_1\pm \:z_2\pm \:z_3\pm ..........\pm \:z_n}\:=\:\overline{z_1}\pm \:\overline{z_2}\:\pm \:\overline{z_3}\pm ..........\pm \overline{z_n}\:}\\\\\mathrm{7.\;\overline{z_1\cdot z_2}\:=\:\overline{z_1}\cdot\overline{z_2}}\\\\\mathrm{\;\;\;In\,\,general,\;\;\overline{z_1\cdot \:z_2\cdot \:z_3\cdot ..........\cdot \:z_n}\:=\:\overline{z_1}\cdot \:\overline{z_2}\:\cdot \:\overline{z_3}\cdot ..........\cdot \overline{z_n}\:}\\\\\mathrm{8.\;\overline{\left(\frac{z_1}{z_2}\right)}=\frac{\overline{z_1}}{\overline{z_2}},\:\:\:z_2\ne 0}\\\\\mathrm{9.\;\;\overline{z^n}\:=\:\left(\overline{z}\right)^n}\\\\\mathrm{10.\;z_1\cdot \overline{z_2}+\overline{z_1}\cdot z_2=2Re\left(z_1\cdot \overline{z_2}\right)\:=\:2Re\left(\overline{z_1}\cdot z_2\right)}$

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