Let's analyze and solve the questions from the image systematically.

Question 1:

Given points $B(1 + i)$ and $C(-2 + 2i)$ in the complex plane.

(a) Find the affix of $A$ such that $ABC$ forms a right isosceles triangle at $A$:

• $A$ must satisfy the condition for the triangle to be right and isosceles, meaning:
  1. $AB = AC$.
  2. The angle at $A$ is $90^\circ$.

The affix $z_A$ of $A$ satisfies: $|z_A - z_B| = |z_A - z_C|$ Using the distances: $\sqrt{(x_A - x_B)^2 + (y_A - y_B)^2} = \sqrt{(x_A - x_C)^2 + (y_A - y_C)^2}.$ Solve for $z_A$.

(b) Construct $ABC$:

• Use the affixes found in part (a) to determine the positions of $A$, $B$, and $C$ on the complex plane.

(c) Find the affix of $E$, symmetric of $A$ with respect to the midpoint of $[BC]$:

• The midpoint of $[BC]$ is: $M = \frac{z_B + z_C}{2}.$ The symmetric point $E$ is: $z_E = 2M - z_A.$

Question 2:

$ABC$ is a right isosceles triangle, direct in $A$. $a$ and $b$ are the affixes of $A$ and $B$.

(a) Find the affix of $C$ in terms of $a$ and $b$:

Since $ABC$ is direct and isosceles, the rotation of $B$ around $A$ by $90^\circ$ gives $C$: $z_C = z_A + i(z_B - z_A).$

(b) Find the affix of $D$ in terms of $a$ and $b$ such that $ABCD$ forms a parallelogram:

In a parallelogram: $z_D = z_C + z_B - z_A.$

Question 3:

Given $A(z_A = 2)$, $B(z_B = 1 + \sqrt{3}i)$, and $Z = \frac{z_B}{z_A}$.

(a) Write $Z$ in exponential form:

$Z = \frac{z_B}{z_A} = \frac{1 + \sqrt{3}i}{2}.$ Convert $Z$ into the exponential form $re^{i\theta}$ by:

1. Finding the modulus $r = |Z|$.
2. Calculating the argument $\theta$ using $\tan^{-1}\left(\frac{\text{Im}(Z)}{\text{Re}(Z)}\right)$.

(b) Deduce the nature of the triangle $OAB$:

• Compute the lengths $OA$, $OB$, and $AB$.
• Use these to classify the triangle $OAB$ (e.g., isosceles, equilateral, or scalene).

Would you like detailed solutions for all parts or a specific question? Let me know! Here are some follow-up questions:

1. How do we verify if $A$ lies on the perpendicular bisector of $BC$?
2. What are the exact coordinates of $C$ in Question 2(a)?
3. How do we visualize the symmetry about the midpoint in Question 1(c)?
4. How can the exponential form in Question 3(a) simplify $Z$?
5. Can we classify $ABCD$ geometrically using complex numbers?

Tip:

Use geometric properties of complex numbers, like rotations and symmetries, to simplify these computations.