If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ and $\mathrm{A}, \mathrm{B}, \mathrm{C} \in R-\{0\}$ such th

Question

If $\mathrm{a}, \mathrm{b}, \mathrm{c}$ and $\mathrm{A}, \mathrm{B}, \mathrm{C} \in R-\{0\}$ such that $a A+b B+c C+\sqrt{\left(a^2+b^2+c^2\right)\left(A^2+B^2+C^2\right)}=0$, then value of $\frac{a B}{b A}+\frac{b C}{c B}+\frac{c A}{a C}$ is

Professor PC Thomas Classes · Answer

Let $r_1=a \hat{i}+\hat{b}+c \hat{k}$ and $r_2=A \hat{i}+B \hat{j}+C \hat{k}$
$\Rightarrow r_1 \cdot r_2=a A+b B+c C$
$\Rightarrow\left|r_1ight|\left|r_2ight|=\sqrt{\left(a^2+b^2+c^2ight)\left(A^2+B^2+C^2ight)}$
$	herefore r_1 r_2=-\left|r_1ight|\left|r_2ight|$
$\Rightarrow r_1$ and $r_2$ are anti-parallel
$\Rightarrow \frac{a}{A}=\frac{b}{B}=\frac{c}{C}=k$, where $k$ is any constant
$
\Rightarrow \frac{a B}{b A}+\frac{C b}{c B}+\frac{c A}{a C}=3
$