Key Concepts and Formulas

• Modulus of a Complex Number: For a complex number $z = x + iy$, the modulus is $|z| = \sqrt{x^2 + y^2}$. Also, $|z|^2 = x^2 + y^2$.
• Properties of Modulus: $\left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|}$ and $|\bar{z}| = |z|$.
• Equation of a Circle: The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$ and the radius is $\sqrt{g^2 + f^2 - c}$. The area of a triangle with vertices $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$ is $\frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$.

Step-by-Step Solution

Step 1: Simplify the given equation

We are given $\left|\frac{\bar{z}-i}{2 \bar{z}+i}\right|=\frac{1}{3}$. Our goal is to manipulate this equation into a recognizable form, specifically the equation of a circle.

Step 1.1: Separate the modulus

Using the property of complex numbers that $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$, we rewrite the equation as: $\frac{|\bar{z}-i|}{|2 \bar{z}+i|} = \frac{1}{3}$

Step 1.2: Cross-multiply to isolate moduli terms

Multiply both sides by $3|2\bar{z}+i|$: $3|\bar{z}-i| = |2 \bar{z}+i|$

Step 2: Convert to Cartesian coordinates

We let $z = x + iy$, so its conjugate is $\bar{z} = x - iy$. Substitute this into the equation: $3|(x-iy)-i| = |2(x-iy)+i|$ $3|x-i(y+1)| = |2x-i(2y-1)|$

Step 3: Square both sides

Squaring both sides eliminates the square roots inherent in the modulus definition: $(3|x-i(y+1)|)^2 = |2x-i(2y-1)|^2$ $9(x^2 + (-(y+1))^2) = (2x)^2 + (-(2y-1))^2$ $9(x^2 + (y+1)^2) = 4x^2 + (2y-1)^2$

Step 4: Expand and simplify

Expand the squared terms: $9(x^2 + y^2 + 2y + 1) = 4x^2 + (4y^2 - 4y + 1)$ $9x^2 + 9y^2 + 18y + 9 = 4x^2 + 4y^2 - 4y + 1$ Now, collect all terms on one side: $5x^2 + 5y^2 + 22y + 8 = 0$ Divide by 5 to get the standard form $x^2+y^2+2gx+2fy+c=0$: $x^2 + y^2 + \frac{22}{5}y + \frac{8}{5} = 0$

Step 5: Identify the center of the circle

The general equation of a circle is $x^2+y^2+2gx+2fy+c=0$, where the center is at $(-g, -f)$. Comparing with our equation: $2g = 0 \implies g = 0$ $2f = \frac{22}{5} \implies f = \frac{11}{5}$ So, the center of the circle $C$ is $(0, -\frac{11}{5})$.

Step 6: Calculate the area of the triangle

We are given that the area of the triangle with vertices $(0,0)$, $C(0, -\frac{11}{5})$, and $(\alpha, 0)$ is 11 square units. The vertices are:

• Origin $O = (0,0)$
• Center $C = (0, -\frac{11}{5})$
• Point $P = (\alpha, 0)$

Step 6.1: Use the triangle area formula

The base of the triangle is the distance from $(0,0)$ to $(\alpha, 0)$, which is $|\alpha|$. The height of the triangle is the distance from the point $C(0, -\frac{11}{5})$ to the x-axis, which is $\frac{11}{5}$. The area of a triangle is given by $\frac{1}{2} \times \text{base} \times \text{height}$. $\text{Area} = \frac{1}{2} \times |\alpha| \times \frac{11}{5}$

Step 6.2: Solve for $\alpha$

We are given that the area is 11 square units: $\frac{1}{2} \times |\alpha| \times \frac{11}{5} = 11$ $\frac{11|\alpha|}{10} = 11$ $|\alpha| = 10$

Step 6.3: Calculate $\alpha^2$

Squaring both sides: $\alpha^2 = (10)^2$ $\alpha^2 = 100$

Common Mistakes & Tips

• Conjugate Confusion: Always remember that if $z = x + iy$, then $\bar{z} = x - iy$.
• Sign Errors: Be meticulous with signs during algebraic manipulations, especially when expanding squared terms.
• Modulus Calculation: Squaring both sides to remove the square root from the modulus simplifies the algebra.

Summary

The problem involves converting a complex number equation into the Cartesian equation of a circle, identifying the circle's center, and then using that center as a vertex of a triangle to find $\alpha^2$, given the triangle's area. After converting the complex equation to Cartesian form and finding the circle's center, the area of the triangle was used to solve for $\alpha^2$. The final calculation for $\alpha^2$ is 100.

Final Answer

The final answer is \boxed{100}, which corresponds to option (B).