Key Concepts and Formulas

• Complex Number Representation: $z = x + iy$, $\bar{z} = x - iy$, $z\bar{z} = x^2 + y^2$
• Circle Equation: $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.
• Distance Formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Step-by-Step Solution

Step 1: Expressing $w$ in terms of $x$ and $y$

We are given $w = z \bar{z} + k_{1} z + k_{2} i z + \lambda(1+i)$, where $k_1, k_2, \lambda \in \mathbb{R}$. We need to express $w$ in terms of its real and imaginary parts by substituting $z = x + iy$. This is essential because the problem provides conditions on $\operatorname{Re}(w)$ and $\operatorname{Im}(w)$.

$w = (x+iy)(x-iy) + k_1(x+iy) + k_2i(x+iy) + \lambda(1+i)$ $w = (x^2 + y^2) + k_1x + ik_1y + ik_2x - k_2y + \lambda + i\lambda$ $w = (x^2 + y^2 + k_1x - k_2y + \lambda) + i(k_1y + k_2x + \lambda)$

Therefore, $\operatorname{Re}(w) = x^2 + y^2 + k_1x - k_2y + \lambda$ $\operatorname{Im}(w) = k_1y + k_2x + \lambda$

Step 2: Determining the equation of circle C

We are given that $\operatorname{Re}(w) = 0$ represents a circle C with radius 1 in the first quadrant, touching the line $y=1$ and the $y$-axis. We need to use this information to find the equation of the circle in standard form.

Since the circle touches the $y$-axis and has radius 1, the $x$-coordinate of the center is 1. Since it touches the line $y=1$ and has radius 1, the $y$-coordinate of the center is $1+1=2$. Thus, the center of the circle is $(1, 2)$ and the radius is 1. The equation of the circle is: $(x-1)^2 + (y-2)^2 = 1^2$ $x^2 - 2x + 1 + y^2 - 4y + 4 = 1$ $x^2 + y^2 - 2x - 4y + 4 = 0$

Step 3: Finding the constants $k_1, k_2, \lambda$

We know that $\operatorname{Re}(w) = 0$ is the equation of circle C. Therefore, we can equate the expression for $\operatorname{Re}(w)$ with the equation of circle C we just found.

$x^2 + y^2 + k_1x - k_2y + \lambda = 0$ $x^2 + y^2 - 2x - 4y + 4 = 0$

Comparing coefficients:

• $k_1 = -2$
• $-k_2 = -4 \implies k_2 = 4$
• $\lambda = 4$

Step 4: Finding the equation of the line $\operatorname{Im}(w) = 0$

Now we substitute the values of $k_1, k_2, \lambda$ into the expression for $\operatorname{Im}(w) = 0$.

$\operatorname{Im}(w) = k_1y + k_2x + \lambda = 0$ $-2y + 4x + 4 = 0$ $4x - 2y + 4 = 0$ $2x - y + 2 = 0$ $y = 2x + 2$

Step 5: Finding the intersection points A and B

To find the intersection points A and B, we need to solve the system of equations formed by the circle and the line.

Substitute $y = 2x + 2$ into the equation of the circle $(x-1)^2 + (y-2)^2 = 1$: $(x-1)^2 + (2x + 2 - 2)^2 = 1$ $(x-1)^2 + (2x)^2 = 1$ $x^2 - 2x + 1 + 4x^2 = 1$ $5x^2 - 2x = 0$ $x(5x - 2) = 0$

This gives us two possible values for $x$: $x = 0$ or $x = \frac{2}{5}$.

Now, find the corresponding $y$ values:

• If $x = 0$, then $y = 2(0) + 2 = 2$. So, point A is $(0, 2)$.
• If $x = \frac{2}{5}$, then $y = 2\left(\frac{2}{5}\right) + 2 = \frac{4}{5} + 2 = \frac{14}{5}$. So, point B is $\left(\frac{2}{5}, \frac{14}{5}\right)$.

Step 6: Calculating the distance $AB$

Now we calculate the distance between A $(0, 2)$ and B $\left(\frac{2}{5}, \frac{14}{5}\right)$: $AB = \sqrt{\left(\frac{2}{5} - 0\right)^2 + \left(\frac{14}{5} - 2\right)^2}$ $AB = \sqrt{\left(\frac{2}{5}\right)^2 + \left(\frac{4}{5}\right)^2}$ $AB = \sqrt{\frac{4}{25} + \frac{16}{25}}$ $AB = \sqrt{\frac{20}{25}} = \sqrt{\frac{4}{5}} = \frac{2}{\sqrt{5}}$ $AB^2 = \frac{4}{5}$

Step 7: Final Calculation: $30(AB)^2$

Finally, we compute the value of $30(AB)^2$: $30(AB)^2 = 30 \times \frac{4}{5} = \frac{120}{5} = 24$

Common Mistakes & Tips

• Careless algebraic manipulation is a common source of error. Double-check each step.
• Visualizing the circle and line in the coordinate plane helps to understand the geometric constraints.
• Remember the relationships between the center, radius, and tangency points of a circle.

Summary

We expressed the complex number $w$ in terms of its real and imaginary parts, used the given information about the circle to determine its equation, solved for the unknown constants, found the equation of the line, determined the intersection points, calculated the distance between the points, and finally computed $30(AB)^2$. The final answer is 24.

Final Answer The final answer is \boxed{24}.