Question

If $(a, b, c)$ & $(a_1, b_1, c_1)$ are two sets of non zero complex numbers satisfying $\Sigma \frac{a}{a_1} = 0$ and $\Sigma \frac{a_1}{a} = 1 - i$ then $\frac{a_1^2}{a^2} + \frac{b^2_1}{b^2} + \frac{c_1^2}{c^2} = $?

• A. $1 + 2i$
• B. $- 2i$
• C. $2i$
• D. $1 - 2i$

Answer 1

Answer: (B)

Given, $\Sigma \frac{a_1}{a} = 1 - i$
$\frac{a_1}{a} + \frac{b_1}{b} + \frac{c_1}{c} = 1 - i \,...(i)$
and $\Sigma \frac{a}{a_1} = 0$
$\Rightarrow \frac{a}{a_1} + \frac{b}{b_1} + \frac{c}{c_1} = 0$
Now squaring (i) both sides we get
$\frac{a_1^2}{a^2} + \frac{b^2_1}{b^2} + \frac{c_1^2}{c_1} = -2i - 2\left(\frac{a_1b_1}{ab} + \frac{b_1c_1}{bc} + \frac{c_1a_1}{ac}\right)$
$= -2i - 2 \frac{a_1b_1c_1}{abc}\left(\frac{c}{c_1} + \frac{a}{a_1} + \frac{b}{b_1}\right) = -2i$