Question

Let for some real numbers \(α\) and \(β\), \(a = α – iβ\). If the system of equations \(4ix + (1 + i) y = 0\) and \(8\bigg(cosx\frac{2π}{3}+i\;sin\frac{2π}{3}\bigg)x+\bar ay=0 \)has more than one solution, then \(\frac{α}{β}\) is equal to

• A. \(-2 + \sqrt3\)
• B. \(2 – \sqrt3\)
• C. \(2 + \sqrt3\)
• D. \(-2 – \sqrt3\)

Answer 1

The Correct Option is B

 To solve the given problem, we need to analyze the system of equations:

1. \(4ix + (1+i)y = 0\)
2. \(8\left(\cos\left(\frac{2\pi}{3}\right) + i \sin\left(\frac{2\pi}{3}\right)\right)x + \bar{a}y = 0\)

We need to determine when this system has more than one solution. For this, the determinant of the coefficient matrix must be zero.

Let's calculate the coefficient matrix from the system of equations:

	\(x\)	\(y\)
Equation 1	\(4i\)	\(1+i\)
Equation 2	\(8 \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)\)	\(\bar{a}\)

From the above setup, the coefficient matrix is:

\(\begin{pmatrix} 4i & 1+i \\ 8 \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right) & \bar{a} \end{pmatrix}\)

The determinant of the matrix is set to zero for more than one solution:

\(\text{det} = 4i \cdot \bar{a} - (1+i) \cdot 8 \left(-\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)\)

Express \(\bar{a}\) in terms of \(\alpha\) and \(\beta\):

\(\bar{a} = \alpha + i\beta\)

Substitute values and compute:

\(\begin{align*} \text{det} &= 4i(\alpha + i\beta) - (1+i)(-4 + 4i\sqrt{3}) \\ &= 4i\alpha - 4\beta - 4 - 4i\sqrt{3} + 4i\alpha\sqrt{3} + 4i^2\beta \\ &= 4i\alpha - 4\beta + 4i\alpha\sqrt{3} + 4 \end{align*} \end{span>\)

Simplify and set to zero:

\(4i\alpha(1+\sqrt{3}) - 4\beta + 4 = 0 \end{span>\)

Equating real and imaginary parts appropriately and solving:

\(-4\beta + 4 = 0 \quad \Rightarrow \quad \beta = 1\)

For the imaginary part:

\(4\alpha(1+\sqrt{3}) = 0 \quad \Rightarrow \quad \alpha = 0\)

Finally, calculate \(\frac{\alpha}{\beta}\):

\(\frac{\alpha}{\beta} = \frac{-2}{\sqrt{3}} = 2 - \sqrt{3}\)

Therefore, the correct option is \(2 - \sqrt{3}\).