## Complex Numbers and Quadratic Equations

# Roots of the Complex Numbers and Geometric Interpretations

**Tips: **

- If z = r (cos θ + i sin θ) and ‘n’ is a positive integer, then \tt z^{\frac{1}{n}} = r^{\frac{1}{n}}\left[\cos\left(\frac{2k\pi + \theta}{n}\right) + i \sin \left(\frac{2k\pi + \theta}{n}\right)\right] where K = 0, 1, 2, 3, ……. (n − 1)
- 1, ω, ω
^{2}are the cube roots of unity - 1 + ω + ω
^{2}= 0 ; ω^{3}= 1, ω^{3n}= 1 ; ω^{3n + 1}= ω ; ω^{3n + 2}= ω^{2} - If a + bω + cω
^{2}= 0, then a = b = c, provided a, b, c are real - ±1, ±i are the fourth roots of unity

**Tricks:**

- Square root of z = a + ib are \tt \pm \left[\sqrt{\frac{|z| + a}{2}} + i \sqrt{\frac{|z| - a}{2}}\right] for b > 0 \tt \pm \left[\sqrt{\frac{|z| + a}{2}} - i \sqrt{\frac{|z| - a}{2}}\right]for b < 0
- \tt \sqrt{z} + \sqrt{\overline{z}} = \pm \sqrt{2}\left\{\sqrt{|z| + a}\right\} where imaginary (z) > 0
- \tt \log i = \log {e}^{\frac{i \pi}{2}} = \frac{i \pi}{2}
- |z – z
_{1}| = |z – z_{2}| ⇒ Locus is perpendicular bisector of the line segment joining z_{1}and z_{2}[z_{1}, z_{2}are fixed points, z is variable] - |z – z
_{1}| + |z – z_{2}| = constant ⇒ Locus of ‘z’ is an ellipse [z_{1}, z_{2}are fixed points, z is variable] - |z – z
_{1}| + |z – z_{2}| = |z_{1}– z_{2}| ⇒ line segment [z_{1}, z_{2}are fixed points, z is variable] - |z – z
_{1}| – |z – z_{2}| = Constant ⇒ Hyperbola [z_{1}, z_{2}are fixed points, z is variable] - |z – z
_{1}| = k |z – z_{2}| (K ≠ 1) ⇒ circle [z_{1}, z_{2}are fixed points, z is variable] - \tt arg \left(\frac{z - z_{1}}{z - z_{2}}\right) = \pm \frac{\pi}{2}\Rightarrow Locus of ‘z’ is a circle with z
_{1}and z_{2}as the vertices of diameter. - \tt arg \left(\frac{z - z_{1}}{z - z_{2}}\right) = o (or) \pi \RightarrowLocus of ‘z’ is a straight line passing through z
_{1}and z_{2} - In general ||z
_{1}| − |z_{2}|| ≤ |z_{1}+ z_{2}| ≤ |z_{1}| + |z_{2}| - If \tt |z + \frac{1}{z}| = a, the greatest and least values of |z| are respectively \tt \frac{a+ \sqrt{a^{2} + 4}}{2} \ and \ \frac{-a + \sqrt{a^{2} + 4}}{2}
- The equation |z – z
_{1}|^{2}+ |z – z_{2}|^{2}= k [k is real number] will represent a circle with centre at \tt \frac{1}{2}(z_{1} + z_{2}) and radius \tt \frac{1}{2}\sqrt{2k - \left(z_{1} - z_{2}\right)^{2}} provided \tt k \geq \frac{1}{2}|z_{1} - z_{2}|^{2} - If z
_{1}, z_{2}, z_{3}…….z_{n}have the same amplititude then |z_{1}| + |z_{2}| + ……… |z_{n}| = |z_{1}+ z_{2}+ z_{3}+ …. + z_{n}| - If ‘3’ points z
_{1}, z_{2}, z_{3}are connected by relation az_{1}+ bz_{2}+ cz_{3}= 0 where a + b + c = 0, then the ‘3’ points are collinear. - If ‘3’ complex number are in A.P. then they lie on a straight line in the complex plane.
- If ‘z’ is a complex number, then e
^{z}is periodic

### Part1: View the Topic in this video From 20:52 To 51:26

### Part2: View the Topic in this video From 00:40 To 52:08

### Part3: View the Topic in this video From 00:40 To 52:20

### Part4: View the Topic in this video From 00:40 To 55:00

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- The area of the triangle whose vertices are z, iz and z + iz is \tt \frac{1}{2} |z|^{2}
- The area of the triangle with vertices z, ωz and z + ωz is \tt \frac{\sqrt{3}}{4} |z|^{2}
- If z
_{1}, z_{2}, z_{3}be the vertices of an equilateral triangle and z_{o}be the circumcentre, then \tt z_{1}^{2} + z_{2}^{2} + z_{3}^{2} = 3z_{0}^{2} - If z
_{1}, z_{2}, z_{3}be the vertices of a triangle, then the triangle is equilateral iff (z_{1}− z_{2})^{2}+ (z_{2}− z_{3})^{2}+ (z_{3}− z_{1})^{2}= 0 (or) \tt z_{1}^{2} + z_{2}^{2} + z_{3}^{3} = z_{1}\cdot z_{2} + z_{2}\cdot z_{3} + z_{3} \cdot z_{1} (or) \frac{1}{z_{1} - z_{2}} + \frac{1}{z_{2} - z_{3}} + \frac{1}{z_{3} - z_{1}} = 0 - x
^{2}+ y^{2}= (x + iy)(x − iy) - x
^{3}+ y^{3}= (x + y)(x + yω)(x + yω^{2}) - x
^{2}+ xy + y^{2}= (x − yω)(x – yω^{2}) in particular x^{2}– x + 1 = (x − ω)(x –ω^{2}) - x
^{2}+ y^{2}+ z^{2}– xy – yz – zx = (x + yω + zω^{2}) (x + yω^{2}+ zω) - x
^{3}+ y^{3}+ z^{3}– 3xyz = (x + y + z) (x + ωy + ω^{2}z) (x + ω^{2}y + ωz) - Equation of the perpendicular bisector is \tt z(\overline{z_{1}} - \overline{z_{2}}) + \overline{z}(z_{1} - z_{2}) = |z_{1}|^{2} - |z_{2}|^{2}
- Equation of a straight line in parametric form z = tz
_{1}+ (1 − t) z_{2}where ‘t’ ∈ R - Non-parametric form is \tt \begin{bmatrix}z & \overline{z} & \overline{1}\\z_{1} & \overline{z_{1}} & 1 \\z_{2} & \overline{z_{2}} & 1 \end{bmatrix} = 0