1. Solve in the set $\mathbb { C }$ of complex numbers the equation (E) with unknown $z$ : $$z ^ { 2 } - 8 z + 64 = 0$$ The complex plane is equipped with a direct orthonormal reference frame $( \mathrm { O } ; \vec { u } , \vec { v } )$. 2. We consider the points $\mathrm { A } , \mathrm { B }$ and C with affixes respectively $a = 4 + 4 \mathrm { i } \sqrt { 3 }$, $b = 4 - 4 \mathrm { i } \sqrt { 3 }$ and $c = 8 \mathrm { i }$. a. Calculate the modulus and an argument of the number $a$. b. Give the exponential form of the numbers $a$ and $b$. c. Show that the points $\mathrm { A } , \mathrm { B }$ and C lie on the same circle with center O whose radius will be determined. d. Place the points $\mathrm { A } , \mathrm { B }$ and C in the reference frame ( $\mathrm { O } ; \vec { u } , \vec { v }$ ). For the rest of the exercise, you may use the figure from question 2. d. completed as the questions progress. 3. We consider the points $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ and $\mathrm { C } ^ { \prime }$ with affixes respectively $a ^ { \prime } = a \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } } , b ^ { \prime } = b \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } }$ and $c ^ { \prime } = c \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } }$. a. Show that $b ^ { \prime } = 8$. b. Calculate the modulus and an argument of the number $a ^ { \prime }$. For the rest we admit that $a ^ { \prime } = - 4 + 4 \mathrm { i } \sqrt { 3 }$ and $c ^ { \prime } = - 4 \sqrt { 3 } + 4 \mathrm { i }$. 4. We admit that if $M$ and $N$ are two points in the plane with affixes respectively $m$ and $n$ then the midpoint $I$ of the segment $[ M N ]$ has affix $\frac { m + n } { 2 }$ and the length $M N$ is equal to $| n - m |$. a. We denote $r , s$ and $t$ the affixes of the midpoints respectively $\mathrm { R } , \mathrm { S }$ and T of the segments $\left[ \mathrm { A } ^ { \prime } \mathrm { B } \right] , \left[ \mathrm { B } ^ { \prime } \mathrm { C } \right]$ and $\left[ \mathrm { C } ^ { \prime } \mathrm { A } \right]$. Calculate $r$ and $s$. We admit that $t = 2 - 2 \sqrt { 3 } + \mathrm { i } ( 2 + 2 \sqrt { 3 } )$. b. What conjecture can be made about the nature of triangle RST? Justify this result.
1. Solve in the set $\mathbb { C }$ of complex numbers the equation (E) with unknown $z$ :
$$z ^ { 2 } - 8 z + 64 = 0$$
The complex plane is equipped with a direct orthonormal reference frame $( \mathrm { O } ; \vec { u } , \vec { v } )$.\\
2. We consider the points $\mathrm { A } , \mathrm { B }$ and C with affixes respectively $a = 4 + 4 \mathrm { i } \sqrt { 3 }$, $b = 4 - 4 \mathrm { i } \sqrt { 3 }$ and $c = 8 \mathrm { i }$.\\
a. Calculate the modulus and an argument of the number $a$.\\
b. Give the exponential form of the numbers $a$ and $b$.\\
c. Show that the points $\mathrm { A } , \mathrm { B }$ and C lie on the same circle with center O whose radius will be determined.\\
d. Place the points $\mathrm { A } , \mathrm { B }$ and C in the reference frame ( $\mathrm { O } ; \vec { u } , \vec { v }$ ).
For the rest of the exercise, you may use the figure from question 2. d. completed as the questions progress.\\
3. We consider the points $\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }$ and $\mathrm { C } ^ { \prime }$ with affixes respectively $a ^ { \prime } = a \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } } , b ^ { \prime } = b \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } }$ and $c ^ { \prime } = c \mathrm { e } ^ { \mathrm { i } \frac { \pi } { 3 } }$.\\
a. Show that $b ^ { \prime } = 8$.\\
b. Calculate the modulus and an argument of the number $a ^ { \prime }$.
For the rest we admit that $a ^ { \prime } = - 4 + 4 \mathrm { i } \sqrt { 3 }$ and $c ^ { \prime } = - 4 \sqrt { 3 } + 4 \mathrm { i }$.\\
4. We admit that if $M$ and $N$ are two points in the plane with affixes respectively $m$ and $n$ then the midpoint $I$ of the segment $[ M N ]$ has affix $\frac { m + n } { 2 }$ and the length $M N$ is equal to $| n - m |$.\\
a. We denote $r , s$ and $t$ the affixes of the midpoints respectively $\mathrm { R } , \mathrm { S }$ and T of the segments $\left[ \mathrm { A } ^ { \prime } \mathrm { B } \right] , \left[ \mathrm { B } ^ { \prime } \mathrm { C } \right]$ and $\left[ \mathrm { C } ^ { \prime } \mathrm { A } \right]$.\\
Calculate $r$ and $s$. We admit that $t = 2 - 2 \sqrt { 3 } + \mathrm { i } ( 2 + 2 \sqrt { 3 } )$.\\
b. What conjecture can be made about the nature of triangle RST? Justify this result.