CAT 2004 — QA Question 10

Given, $\frac{\mathrm{a}}{\mathrm{b}+\mathrm{c}}=\frac{\mathrm{b}}{\mathrm{c}+\mathrm{a}}=\frac{\mathrm{c}}{\mathrm{c}+\mathrm{a}}=\mathrm{r}$

By property of equal ratios,

$\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{\mathrm{b}+\mathrm{c}+\mathrm{c}+\mathrm{a}+\mathrm{a}+\mathrm{b}}$ = r

∴ $\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{2(\mathrm{a}+\mathrm{b}+\mathrm{c})}$ = r

Assuming (a + b + c ≠ 0),

∴ r = $\frac{1}{2}$

If a + b + c = 0, a = –(b + c)

Then from the original ratio, r = $\frac{\mathrm{a}}{\mathrm{b}+\mathrm{c}}$ = $\frac{-(\mathrm{b}+\mathrm{c})}{(\mathrm{b}+\mathrm{c})}$= -1

Hence, option (c).