# JEE Geometry of Complex Numbers

This page will teach you how to master JEE Geometry of Complex Numbers. We highlight the main concepts, provide a list of examples with solutions, and include problems for you to try. Once you are confident, you can take the quiz to establish your mastery.

## JEE conceptual theory

As per the JEE syllabus, the main concepts under Geometry of Complex Numbers are geometrical formulas, conditions for quadrilaterals, straight line in an argand plane, circle in an argand plane, and some other important loci in an argand plane. Some questions that are hard in cartesian plane can be solved easily by argand plane like the problems of rotating a point by 90 degree anticlockwise about origin is same as multiplying \(i\) to that point in argand plane, and you get the point easily.

What are the coordinates of point when the point \((1,2)\) is rotated about origin \(90^{\circ}\) anticlockwise?

Convert into argand plane to get the point as \(1+2i\). Now according to the point written above, multiply by \(i\) to get the required answer as \(-2+i\). Return back to cartesian plane. Thus the answer is \((-2,1)\)

### Geometrical formulas

Section formula

Different centers of a triangle

### Conditions for quadrilaterals

Parallelogram: \(z_1+z_3=z_2+z_4\)

Rectangle: \(z_1+z_3=z_2+z_4\) and \(|z_3-z_1|=|z_4-z_2|\)

Rhombus: \(z_1+z_3=z_2+z_4\) and \(|z_4-z_1|=|z_2-z_1|\)

Square: \(z_1+z_3=z_2+z_4\),\(|z_4-z_1|=|z_2-z_1|\) and \(|z_2-z_1|=|z_4-z_2|\)

### Straight line in an argand plane

- Parametric equation: \(z=tz_1+(1-t)z_2\),\(0 \le t \le 1\) \(t\) being a real parameter

The parametric equation is based on the fact that as \(t\) varies from \(0\) to \(1\) the variable complex number \(z\) changes from \(z_2\) to \(z_1\) tracing all the points between the complex numbers \(z_1\) and \(z_2\) lying on the line \((z_1-z_2)\). So, rather than a line it represents a line segment.

- Slope and complex slope: complex slope of the line \(\overline{\alpha}z+\alpha \overline z +c=0\) is \(-\left( \frac{\alpha}{\overline{\alpha}} \right)\).

Real slope of the line : \(-\frac{\text{Re(a)}}{\text{Im(a)}}\)

- Image of a point with respect to a straight line

### Circle in an Argand plane

**Equations of a circle:-**

Parametric equation of a circle:- If the the equation of a circle is represented by \(\left| z-a\right|=b\), then its parametric equation is \(z=a+b{e}^{i\theta}, 0 \le \theta \le 2\pi\)

Equation of a circle passing through three non collinear points \(z_1, z_2, z_3\) is given by

\(\left( \dfrac{z-z_1}{\bar{z}-\bar{z_1}}\right)\left( \dfrac{z_2-z_3}{\bar{z_2}-\bar{z_3}} \right)=\left( \dfrac{z-z_2}{\bar{z}-\bar{z_2}}\right)\left( \dfrac{z_1-z_3}{\bar{z_1}-\bar{z_3}} \right)\)

- Equation of the circle whose one diameter is the line segment joining \(z_1\) and \(z_2\) is given by

\(\dfrac{z-z_1}{z-z_2}+\dfrac{\bar{z}-\bar{z_1}}{\bar{z}-\bar{z_2}}=0\)

- General equation of a circle is given as \(\bar{z}z+\bar{a}z+a\bar{z}+b=0\), where \(b\) is real. The centre of this circle is \(-a\) and radius is \(\sqrt{a\bar{a}-b}\)

### Some other important loci in an argand plane are listed below.

\(|z-z_1|=|z-z_2|\): Locus of \(z\) is perpendicular bisector of the line segment joining points \(z_1\) and \(z_2\).

\(|z-z_1|+|z-z_2|=|z_1-z_2|\): Locus of \(z\) is the line segment joining \(z_1\) and \(z_2\).

\(\left| |z-z_1|-|z-z_2| \right|=|z_1-z_2|\): Locus of \(z\) is a straight line joining \(z_1\) and \(z_2\) but \(z\) does not lie between \(z_1\) and \(z_2\).

\(|z-z_1|+|z-z_2|=2a\), where \(2a>|z_1-z_2|\): Locus of \(z\) is an ellipse having foci at \(z_1\) and \(z_2\).

\(\left| |z-z_1|-|z-z_2| \right|=2a\), where \(2a<|z_1-z_2|\) and \(a\in \mathbb R^+\): Locus of \(z\) is an hyperbola having foci at \(z_1\) and \(z_2\).

\(\left| \frac{z-z_1}{z-z_2} \right|=k\,(k>0, k \neq 1)\): Locus of \(z\) is a circle.

\(|z-z_1|^2+|z-z_2|^2=k\): Locus of \(z\) is a circle provide \( k \geq \frac 12 |z_1-z_2|^2\).

## JEE main problems

\[ \begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array} \]

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\[ \begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array} \]

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## JEE advanced problems

\[ \begin{array} { l l } A) \, & \quad \quad \quad \quad \quad & B) \, \\ C) \, & & D) \, \\ \end{array} \]

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Once you are confident of Complex Numbers, move on to JEE Complex Numbers - Advanced Conceptual Understanding.

**Cite as:**JEE Geometry of Complex Numbers.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/jee-geometry-complex-numbers/