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Conjugate of complex numbers and their properties - (Concept)
The conjugate of a complex number $z=a+i b(a, b$ are real numbers) is $a-i b$. It is denoted as $\bar{z}$. i.e. if $z=a+i b$, then its conjugate is $\bar{z}=a-i b$.
The 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. $\mathrm{z}=\mathrm{a}+$ $\mathrm{ib}, \bar{z}=\mathrm{a}-\mathrm{ib}$
Then the sum, $z+\bar{z}=a+i b+a-i b=2 a$ (which is real)
When a complex number is multiplied by its complex conjugate, the result is a real number i.e. $z=$ $\mathrm{a}+\mathrm{ib}, \bar{z}=\mathrm{a}-\mathrm{ib}$
Then the product, $z \cdot \bar{z}=(a+i b) \cdot(a-i b)=a^2-(i b)^2$
$
=a^2+b^2(\text { which is real })
$
Geometrically complex conjugate of a complex number is its mirror image with respect to the real axis (x-axis).
For example
$
\mathrm{z}=2+2 \mathrm{i} \text { and } \bar{z}=2-2 i
$
Properties of the conjugate complex numbers:
z, z1, z2, and z3 be the complex numbers
1. $\overline{(\bar{z})}=z$
2. $\mathrm{z}+\overline{\mathrm{z}}=2 \cdot \operatorname{Re}(\mathrm{z})$
3. $\mathrm{z}-\overline{\mathrm{z}}=2 \mathrm{i} \cdot \operatorname{Im}(\mathrm{z})$
4. $\mathrm{z}+\overline{\mathrm{z}}=0 \Rightarrow \mathrm{z}=-\overline{\mathrm{z}} \Rightarrow \mathrm{z}$ is purely imaginary
$5 . \mathrm{z}-\overline{\mathrm{z}}=0 \Rightarrow \mathrm{z}=\overline{\mathrm{z}} \Rightarrow \mathrm{z}$ is purely real
6. $\overline{z_1 \pm z_2}=\overline{z_1} \pm \overline{z_2}$
In general, $\overline{z_1 \pm z_2 \pm z_3 \pm \ldots \ldots \ldots \pm \mathrm{z}_n}=\overline{z_1} \pm \overline{z_2} \pm \overline{z_3} \pm \ldots \ldots \ldots \pm \overline{z_n}$
7. $\overline{\mathrm{z}_1 \cdot \mathrm{Z}_2}=\overline{\mathrm{z}_1} \cdot \overline{\mathrm{z}_2}$
In general, $\overline{z_1 \cdot z_2 \cdot z_3 \cdot \ldots \ldots \ldots \cdot \cdot} \overline{z_n}=\overline{z_1} \cdot \overline{z_2} \cdot \overline{z_3} \cdot \ldots \ldots \ldots \cdot \overline{z_n}$
8. $\overline{\left(\frac{z_1}{z_2}\right)}=\frac{\overline{z_1}}{\overline{z_2}}, \quad z_2 \neq 0$
9. $\overline{\mathrm{z}^{\mathrm{n}}}=(\overline{\mathrm{z}})^{\mathrm{n}}$
10. $\mathrm{z}_1 \cdot \overline{\mathrm{z}_2}+\overline{\mathrm{z}_1} \cdot \mathrm{z}_2=2 \operatorname{Re}\left(\mathrm{z}_1 \cdot \overline{\mathrm{z}_2}\right)=2 \operatorname{Re}\left(\overline{\mathrm{z}_1} \cdot \mathrm{z}_2\right)$
| Exam | Chapter |
| JEE MAIN | Complex numbers and quadratic equations |
If , where
then,
=
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Conjugate of will be $\frac{x}{10}+\frac{y}{10}i$, then find the value of $x+y$.
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z is a complex number such that and
then z equals
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z is a complex number such that , then
equals
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Let &
are two complex numbers such that
then arg
equals $\frac{-\pi}{n}$, then find the value of $n$.
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Let z is a complex number with its conjugate as then
equals
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If $\frac{z-\alpha }{z+\alpha }\left ( \alpha \in {R} \right )$ is a purely imaginary number and $\left | z \right |=2$, then the positive value of is :
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If then
equals
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The value of is
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Let the lines $(2-i) \bar{z}=(2+i) \bar{z}-4 i=0,\left(\text { here } i^2=-1\right) {\text {be normal }}$ to a circle C . If the line $i z+\bar{z}+1+i=0$ is tangent to the circle C , then its radius is:
| A. |
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| C. |
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| D. |
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Let and
are two complex numbers such that
then
equals
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| D. |
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If is conjugate of
then
equals
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The conjugate of a complex number is . Then that complex number is:
| A. |
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| D. |
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Let z be a complex number such that then
equals
| A. |
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| D. |
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Let z & w are complex numbers satisfying then
equals
| A. |
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| D. |
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If then
equals
| A. |
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| D. |
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Sum of squares of modulus of all the complex numbers satisfying
is equal to ___________.
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| D. |
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Let . Then
is equal to ____________.
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Let be the set of all
, for which the complex number
is purely imaginary and
is purely real. Let
. Then
is equal to :
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| C. |
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| D. |
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Let be complex numbers satisfying
Then the least value of
such that
is equal to ____________.
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| D. |
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if then
is equal to :
| A. |
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| D. |
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The conjugate of , if
, is.
| A. |
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The conjugate of is.
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Let be two-zero real numbers. Then the number of elements in the set
and
is equal to :
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Let . Then which of the following is NOT correct?
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Which one of the following substances has the highest proton affinity?
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If z1, z2 and z3, z4 are 2 pairs of complex conjugate numbers, thenequals:
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Let $\alpha$ and $\beta$ be the sum and the product of all the non-zero solutions of the equation $(\bar{z})^2+|z|=0, z \in C$. Then $4\left(\alpha^2+\beta^2\right)$ is equal to:
| A. |
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| D. |
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If $\alpha$ and $\beta$ are different complex numbers with $|\beta|=1$, then $\left|\frac{\beta+\alpha}{1+\bar{\alpha} \beta}\right|_{\text {is equal to }}$
| A. |
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If $f(z)=0$ has $\bar{z}_1$ and $\bar{z}_2$ as its solution, then $f(\bar{z})=0$ has its solution
| A. |
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| D. |
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Match the column
$z$
(i) $2+3 i$
(ii) $i$
(iii) 4
(iv) $-1+i$
and
$\bar{z}$
$
(p)-1-i
$
$(q) 4$
$
\begin{aligned}
& (r)-i \\
& (s) 2-3 i
\end{aligned}
$
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What is conjugate of product of two complex no's, Whose product of conjugates is non-zero purely imaginary?
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| D. |
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$
\text { What is the value of } \overline{(\bar{z})} \text { if } \bar{z}=3+4 i ?
$
| A. |
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| C. |
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| D. |
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If $z_1=2-3 i$ and $z_2=-1+i$, then $\overline{z_1+z_2}=$
| A. |
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| D. |
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If $z_1=41+i$ and $z_2=-39-4 i$ then $\overline{z_1-z_2}$ is
| A. |
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| C. |
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| D. |
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What is the value of $\left(\frac{1+i}{-2-3 i}\right)$
| A. |
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| D. |
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What is $\operatorname{Im}(z)$, so that $z-\bar{z}=-6 i$
| A. |
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| C. |
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| D. |
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If $i=\sqrt{-1}$ then $4+5\left(\frac{-1}{2}+\frac{i \sqrt{3}}{2}\right)^{334}-3\left(\frac{1}{2}+\frac{i \sqrt{3}}{2}\right)^{365}$ is equal to:
| A. |
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| B. |
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| C. |
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| D. |
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If the conjugate of $i+1$ is $z_1$, the conjugate of $-4 i$ is $z_2$ and conjugate of 3 is $z_3$, then $z_1+z_2+z_3$ is
| A. |
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| B. |
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| C. |
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| D. |
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Let $\mathrm{z}$ be a complex number such that the real part of $\frac{z-2 i}{z+2 i}$ is zero. Then, the maximum value of $|\mathrm{z}-(6+8 \mathrm{i})|$ is equal to :
| A. |
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| B. |
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| D. |
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If z = x + iy lies in the third quadrant, the $\frac{\bar{z}}{z}$ also lies in the third quadrant if
| A. |
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| C. |
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| D. |
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Find $\bar{z}^{\bar{n}}=$ ?, if $\bar{z}=1-4 i \ and \ \ n=2$
| A. |
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| B. |
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| C. |
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| D. |
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What is $\frac{\overline{z_1}}{\bar{z}_2}, i f\left(\frac{\bar{z}_1}{\bar{z}_2}\right)=3-4 i$
| A. |
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| B. |
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| C. |
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| D. |
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Let $z\:(\neq1)$ be a complex number such that |z|=1 imaginary part of $\frac{\bar{z}(1-z)}{z(1+\bar{z})}$ ?
| A. |
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| B. |
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| C. |
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| D. |
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If $\arg (Z)=0$, then $Z-\bar{Z}=$
| A. |
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| B. |
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| C. |
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| D. |
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What is $\operatorname{Re}(z)$, such that $z+\bar{z}=8$
| A. |
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| B. |
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| C. |
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| D. |
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Let z = a + ib be a complex number. the conjugate of z, denoted as , is the complex number a – ib, i.e.,
= a – ib.
