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Distance formula, Equation of perpendicular bisector - (Concept)
Distance between two points $\mathrm{A}\left(\mathrm{z}_1\right)$ and $\mathrm{B}\left(\mathrm{z}_2\right)$ is
$
A B=\left|z_2-z_1\right|=\mid \text { Affix of } B-\text { Affix of } A \mid
$
Let $\mathrm{z}_1=\mathrm{x}+\mathrm{iy}$ and $\mathrm{z}_2=\mathrm{x}+\mathrm{iy}$ Then,$\left|z_1-z_2\right|=\left|\left(x_1-x_2\right)+i\left(y_1-y_2\right)\right|$$=\sqrt{\left(\mathrm{x}_1-\mathrm{x}_2\right)^2+\left(\mathrm{y}_1-\mathrm{y}_2\right)^2}$$=$ distance between points $\left(\mathrm{x}_1, \mathrm{y}_1\right)$ and $\left(\mathrm{x}_2, \mathrm{y}_2\right)=$ distance between between $\mathrm{z}_1$ and $\mathrm{z}_2$ where $\mathrm{z}_1=\mathrm{x}_1+\mathrm{iy}_1$ and $\mathrm{z}_2=\mathrm{x}_2+\mathrm{iy}_2$
The distance of a point from the origin is $|z-0|=|z|$
Three points $A\left(z_1\right), B\left(z_2\right)$ and $C\left(z_3\right)$ are collinear, then $A B+B C=A C$
i.e. $\left|z_2-z_1\right|+\left|z_3-z_2\right|=\left|z_3-z_1\right|$
Perpendicular bisector
We can use the distance formula to find the equation of perpendicular bisector
Let two fixed points A(z1) and B(z2) and a moving point C(z) which lies on the perpendicular bisector of AB
As any point on the perpendicular bisector of AB will be equidistant from A and B, so
$
\begin{aligned}
& A C=B C \\
& \left|z-z_1\right|=\left|z-z_2\right|
\end{aligned}
$
This is the equation of perpendicular to the bisector of $A B$, where $A\left(z_1\right)$ and $B\left(z_2\right)$.
| Exam | Chapter |
| JEE MAIN | Complex numbers and quadratic equations |
Distance (in units) between and
equals
| A. |
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| B. |
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| C. |
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| D. |
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If and z moves such that
then
equals
| A. |
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| B. |
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| C. |
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| D. |
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If then locus of z will be
| A. |
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| B. |
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| D. |
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The largest value of r for which the region represented by the set $\{w \epsilon C /|w-4-i| \leq r\}$ is contained in the region represented by the set $\{z \epsilon C /|z-1| \leq|z+i|\}$ is equal to:
| A. |
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| B. |
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| C. |
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| D. |
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The number of complex numbers $z$ such that $|z-1|=|z+1|=|z-i| {\text { equals }}$
| A. |
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| B. |
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| C. |
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| D. |
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Let $z \varepsilon C$ be such that $|z|<1$. If $\omega=\frac{5+3 z}{5(1-z)}$, then :
| A. |
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| B. |
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| C. |
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| D. |
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Let $z$ be a complex number such that $\left|\frac{z-i}{z+2 i}\right|=1 \quad|z|=\frac{5}{2}$. Then the value of $|z+3 i|_{\text {is : }}$
| A. |
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| B. |
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| C. |
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| D. |
|
Let $z$ and $w$ be two complex numbers such that
$
w=z \bar{z}-2 z+2,\left|\frac{z+i}{z-3 i}\right|=1
$
and $\mathrm{Re}(w)$ has minimum value. Then, the minimum value of $n \in N$ for which $w^n$ is real, is equal to $\qquad$
| A. |
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| B. |
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| C. |
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| D. |
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If and z moves such that
, then
equals
| A. |
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| B. |
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| C. |
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| D. |
|
If &
are equidistant from
then maximum value of 'a' equals
| A. |
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| B. |
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| C. |
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| D. |
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If $|z-4|<|z-2|$, its solution is given by
| A. |
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| B. |
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| C. |
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| D. |
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Let $\mathrm{A}=\left\{z \in \mathbf{C}:\left|\frac{z+1}{z-1}\right|<1\right\}$ and $\mathrm{B}=\left\{z \in \mathbf{C}: \arg \left(\frac{z-1}{z+1}\right)=\frac{2 \pi}{3}\right\}$ Then $A \cap B$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of points of intersection of $\mathrm{|z-(4+3 i)|=2\; and \;|z|+|z-4|=6}$, $\mathrm{z \in \mathbb{C}}$, is
| A. |
|
| B. |
|
| C. |
|
| D. |
|
The number of elements in the set
$\mathrm{\{z=a+i b \in \mathbb{C}: a, b \in \mathbb{Z}\; and\; \mathrm{1<|z-3+2 i|<4\}}}$ is
| A. |
|
| B. |
|
| C. |
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| D. |
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Let the minimum value is attained at
is equal to
| A. |
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| B. |
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| C. |
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| D. |
|
Let $\mathbb{C}$ be the set of all complex numbers. Let $S_1=\{z \in \mathbb{C}:|z-2| \leq 1\}$ and $S_2=\{\mathrm{z} \in \mathbb{C}: \mathrm{z}(1+\mathrm{i})+\overline{\mathrm{z}}(1-\mathrm{i}) \geq 4\}$
Then, the maximum value of $\left|z-\frac{5}{2}\right|^2$ for $\mathrm{z} \in \mathrm{S}_1 \cap \mathrm{~S}_2$ is equal to
| A. |
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| B. |
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| C. |
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| D. |
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If , then :
| A. |
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| B. |
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| C. |
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| D. |
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Let and
.The set
represents a
| A. |
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| B. |
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| C. |
|
| D. |
|
If the set is equal to the interval
, then
is equal to
| A. |
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| B. |
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| C. |
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| D. |
|
If $\alpha$ denotes the number of solutions of $|1-i|^x=2^x$ and $\beta=\left(\frac{|z|}{\arg (z)}\right)$, where $z=\frac{\pi}{4}(1+i)^4\left[\frac{1-\sqrt{\pi} i}{\sqrt{\pi}+i}+\frac{\sqrt{\pi}-i}{1+\sqrt{\pi} i}\right], i=\sqrt{-1}$, then the distance of the point $(\alpha, \beta)$ from the line $4 x-3 y=7$ is_______.
| A. |
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| B. |
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| C. |
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| D. |
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What is the length of the perpendicular foot drawn from point P (3, 4, 5) on the y-axis?
| A. |
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| B. |
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| C. |
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| D. |
|
Let $\left|\mathrm{z}_1-8-2 \mathrm{i}\right| \leq 1 $ and $\quad\left|\mathrm{z}_2-2+6 \mathrm{i}\right| \leq 2$, $z_1, z_2 \in C$. Then the minimum value of $\left|z_1-z_2\right|$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
Complex number that divides line segment joining $z_1=(1+i)$ and $z_2=-1$ internally in ratio $1: 2$ is
| A. |
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| B. |
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| C. |
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| D. |
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If $z=x+i y$, then the equation $|z+1|=|z-1| {\text {represents }}$
| A. |
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| B. |
|
| C. |
|
| D. |
|
If $Z_1=2-3 i$ and $Z_2=-1+i$, then distance between $Z_{1}$ and $ Z_2$ is:
| A. |
|
| B. |
|
| C. |
|
| D. |
|
