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

Answer: (B)

$2 – \sqrt3$

Answer 2

Given $a = α – iβ$ and

$4ix \+ (1 + i)y = 0 …(i)$

$8\bigg(cosx\frac{2π}{3}+i\;sin\frac{2π}{3}\bigg)x+\bar ay=0....(ii)$

By (i) $\frac{x}{y}=\frac{−(1+i)}{4i}$…(iii)

By (ii) $\frac{x}{y}=\frac{−\bar a}{8\bigg(\frac{−1}{2}+\frac{\sqrt{3t}}{2}\bigg)}$….(iv)

Now by $(iii)$ and $(iv)$

$\frac{1+i}{4i}=\frac{\bar a}{4(−1+\sqrt3i)}$

$⇒\bar a=(\sqrt3–1)+(\sqrt3+1)i$

$⇒α+iβ=(\sqrt3–1)+(\sqrt3+1)i$

$∴\frac{α}{β}=\frac{\sqrt3−1}{\sqrt3+1}=2–\sqrt3$