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Sagot :
To solve this problem, we need to follow the provided steps and explanations:
### (1) Determining [tex]\( z_3 \)[/tex] in the form of [tex]\( a + bi \)[/tex]
Given:
[tex]\[ z_1 = 2 + i \][/tex]
[tex]\[ z_2 = -2 + 4i \][/tex]
We are required to find:
[tex]\[ z_3 = \frac{1}{z_1} + \frac{1}{z_2} \][/tex]
To find [tex]\(\frac{1}{z_1}\)[/tex] and [tex]\(\frac{1}{z_2}\)[/tex], we use the property of complex conjugates. For a complex number [tex]\( z = a + bi \)[/tex], the reciprocal is given by:
[tex]\[ \frac{1}{z} = \frac{\bar{z}}{z\bar{z}} = \frac{a - bi}{a^2 + b^2} \][/tex]
So, for [tex]\( z_1 = 2 + i \)[/tex]:
[tex]\[ \frac{1}{z_1} = \frac{2 - i}{2^2 + 1^2} = \frac{2 - i}{5} = \frac{2}{5} - \frac{1}{5}i \][/tex]
For [tex]\( z_2 = -2 + 4i \)[/tex]:
[tex]\[ \frac{1}{z_2} = \frac{-2 - 4i}{(-2)^2 + (4i)^2} = \frac{-2 - 4i}{4 + 16} = \frac{-2 - 4i}{20} = -\frac{1}{10} - \frac{2}{5}i \][/tex]
Now, summing these results to find [tex]\( z_3 \)[/tex]:
[tex]\[ z_3 = \frac{1}{z_1} + \frac{1}{z_2} = \left( \frac{2}{5} - \frac{1}{5}i \right) + \left( -\frac{1}{10} - \frac{2}{5}i \right) \][/tex]
Combining real and imaginary parts:
[tex]\[ \text{Real part} = \frac{2}{5} - \frac{1}{10} = \frac{4}{10} - \frac{1}{10} = \frac{3}{10} = 0.3 \][/tex]
[tex]\[ \text{Imaginary part} = -\frac{1}{5} - \frac{2}{5} = -\frac{1}{5} \left( 1 + 2 \right) = -\frac{3}{5} = -0.6 \][/tex]
So:
[tex]\[ z_3 = 0.3 - 0.6i \][/tex]
### (ii) Representing [tex]\( z_1z_2 \)[/tex] and [tex]\( z_3 \)[/tex] in the Argand diagram
- Calculating [tex]\( z_1z_2 \)[/tex]:
[tex]\[ z_1 = 2 + i \][/tex]
[tex]\[ z_2 = -2 + 4i \][/tex]
To find [tex]\( z_1z_2 \)[/tex]:
[tex]\[ z_1z_2 = (2 + i)(-2 + 4i) \][/tex]
Using the distributive property:
[tex]\[ z_1z_2 = 2(-2 + 4i) + i(-2 + 4i) \][/tex]
[tex]\[ z_1z_2 = -4 + 8i - 2i + 4i^2 \][/tex]
[tex]\[ z_1z_2 = -4 + 6i + 4(-1) \][/tex]
[tex]\[ z_1z_2 = -4 + 6i - 4 \][/tex]
[tex]\[ z_1z_2 = -8 + 6i \][/tex]
So, [tex]\( z_1z_2 = -8 + 6i \)[/tex].
To represent these complex numbers on the Argand diagram, we plot the following points:
- [tex]\( z_1z_2 = -8 + 6i \)[/tex] is represented by the point (-8, 6).
- [tex]\( z_3 = 0.3 - 0.6i \)[/tex] is represented by the point (0.3, -0.6).
These would be plotted as corresponding points in the complex plane with their real parts on the x-axis and their imaginary parts on the y-axis.
### (1) Determining [tex]\( z_3 \)[/tex] in the form of [tex]\( a + bi \)[/tex]
Given:
[tex]\[ z_1 = 2 + i \][/tex]
[tex]\[ z_2 = -2 + 4i \][/tex]
We are required to find:
[tex]\[ z_3 = \frac{1}{z_1} + \frac{1}{z_2} \][/tex]
To find [tex]\(\frac{1}{z_1}\)[/tex] and [tex]\(\frac{1}{z_2}\)[/tex], we use the property of complex conjugates. For a complex number [tex]\( z = a + bi \)[/tex], the reciprocal is given by:
[tex]\[ \frac{1}{z} = \frac{\bar{z}}{z\bar{z}} = \frac{a - bi}{a^2 + b^2} \][/tex]
So, for [tex]\( z_1 = 2 + i \)[/tex]:
[tex]\[ \frac{1}{z_1} = \frac{2 - i}{2^2 + 1^2} = \frac{2 - i}{5} = \frac{2}{5} - \frac{1}{5}i \][/tex]
For [tex]\( z_2 = -2 + 4i \)[/tex]:
[tex]\[ \frac{1}{z_2} = \frac{-2 - 4i}{(-2)^2 + (4i)^2} = \frac{-2 - 4i}{4 + 16} = \frac{-2 - 4i}{20} = -\frac{1}{10} - \frac{2}{5}i \][/tex]
Now, summing these results to find [tex]\( z_3 \)[/tex]:
[tex]\[ z_3 = \frac{1}{z_1} + \frac{1}{z_2} = \left( \frac{2}{5} - \frac{1}{5}i \right) + \left( -\frac{1}{10} - \frac{2}{5}i \right) \][/tex]
Combining real and imaginary parts:
[tex]\[ \text{Real part} = \frac{2}{5} - \frac{1}{10} = \frac{4}{10} - \frac{1}{10} = \frac{3}{10} = 0.3 \][/tex]
[tex]\[ \text{Imaginary part} = -\frac{1}{5} - \frac{2}{5} = -\frac{1}{5} \left( 1 + 2 \right) = -\frac{3}{5} = -0.6 \][/tex]
So:
[tex]\[ z_3 = 0.3 - 0.6i \][/tex]
### (ii) Representing [tex]\( z_1z_2 \)[/tex] and [tex]\( z_3 \)[/tex] in the Argand diagram
- Calculating [tex]\( z_1z_2 \)[/tex]:
[tex]\[ z_1 = 2 + i \][/tex]
[tex]\[ z_2 = -2 + 4i \][/tex]
To find [tex]\( z_1z_2 \)[/tex]:
[tex]\[ z_1z_2 = (2 + i)(-2 + 4i) \][/tex]
Using the distributive property:
[tex]\[ z_1z_2 = 2(-2 + 4i) + i(-2 + 4i) \][/tex]
[tex]\[ z_1z_2 = -4 + 8i - 2i + 4i^2 \][/tex]
[tex]\[ z_1z_2 = -4 + 6i + 4(-1) \][/tex]
[tex]\[ z_1z_2 = -4 + 6i - 4 \][/tex]
[tex]\[ z_1z_2 = -8 + 6i \][/tex]
So, [tex]\( z_1z_2 = -8 + 6i \)[/tex].
To represent these complex numbers on the Argand diagram, we plot the following points:
- [tex]\( z_1z_2 = -8 + 6i \)[/tex] is represented by the point (-8, 6).
- [tex]\( z_3 = 0.3 - 0.6i \)[/tex] is represented by the point (0.3, -0.6).
These would be plotted as corresponding points in the complex plane with their real parts on the x-axis and their imaginary parts on the y-axis.
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