**Equality:** Two complex numbers are equal **if and only if** their real parts and their imaginary parts are respectively equal.

a+ib =c+id if and only if a=c and b=d

**Example 1:** If *x+4yi+40 = 5x+8i*, Find the values x and y.

__Solution__**: **Equating the** **left hand side of the real part = Right hand side of the real part

*x +40 = 5x
*

*5x-x = 40
*

*4x=40
*

*x=10
*

Left hand side of the imaginary part = Right hand side of the imaginary part

*4y = 8
*

*y = 2
*

Therefore the value x is 10 and the value y is 2.

**Example 2: **Find the values of a and b for the following equations:

**Solution: **

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**Example 3: **If z = x + iy and |2z- 1| = |z- 2| then prove that x^{2} + y^{2} = 1.

__Solution __:

|2z- 1| = |z- 2|

→|2(x+iy)- 1| = |x+iy- 2|

→|(2x-1)+2iy| = |(x-2)+iy|

→√(2x − 1) 2 + 4y2 = √(x − 2)2+ y2

→4x2 - 4x + 1 + 4y2 = x2 − 4x + 4 + y2

(squaring on both sides)

→3x2 + 3y2 - 3 = 0

→3x2 + 3y2 = 3

→x2 + y2 = 1. (Proved).**
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**Example 4:**** Find the real numbers a and b such that the equation below becomes true.
**

**(a + 6) + 2bi = 4 - 5i
**

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__Solution__**: ****(a + 6) + 2bi = 4 - 5i
**

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Equating the real and imaginary parts, we get,

a + 6 = 4 and 2b = -5

a = -2 and b = -5/2