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 $\frac{a_1^2}{a^2}+\frac{b_1^2}{b^2}+\frac{c_1^2}{c^2}=$?

• 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
$\frac{a_1^2}{a^2}+\frac{b_1^2}{b^2}+\frac{c_1^2}{c_1}=-2i-2\left(\frac{a_1{b}_1}{ab}+\frac{b_1{c}_1}{bc}+\frac{c_1{a}_1}{ac}\right)$
$=-2i-2\frac{a_1{b}_1{c}_1}{abc}\left(\frac{c}{c_1}+\frac{a}{a_1}+\frac{b}{b_1}\right)=-2i$