Question

If $a,b,c,p,q,r$ are three complex numbers such that $\frac{p}{a} + \frac{q}{b} + \frac{r}{c} = 1 + i$and $\frac{a}{p} + \ \frac{b}{q} + \frac{c}{r} = 0$ then value of $\frac{p^{2}}{a^{2}} + \frac{q^{2}}{b^{2}} + \frac{r^{2}}{c^{2}}$is

• A. 0
• B. -1
• C. 2i
• D. -2i

Answer 1

Answer: (C)

2i

Answer 2

Sol. $\frac{p}{a} + \frac{q}{b} + \frac{r}{c} = 1 + i,$

⇒ $\left( \frac{p}{a} + \frac{q}{b} + \frac{r}{c} \right)^{2} = (1 + i)^{2} = 2i$

⇒ $\frac{p^{2}}{a^{2}} + \frac{q^{2}}{b^{2}} + \frac{r^{2}}{c^{2}} + 2\left( \frac{qr}{bc} + \frac{rp}{ca} + \frac{pq}{ab} \right) = 2i$

⇒ $\frac{p^{2}}{a^{2}} + \frac{q^{2}}{b^{2}} + \frac{r^{2}}{c^{2}} + \frac{2abc}{pqr}\left( \frac{a}{p} + \frac{b}{q} + \frac{c}{r} \right) = 2i$

⇒ $\frac{p^{2}}{a^{2}} + \frac{q^{2}}{b^{2}} + \frac{r^{2}}{c^{2}} = 2i$