If the equations $2{\mathrm{ax}}^2-3\mathrm{bx}+4\mathrm{c}=0$ and  $3{\mathrm{x}}^2-4\mathrm{x}+5=0$  have a common root, then $(\mathrm{a}+\mathrm{b})/(\mathrm{b}+\mathrm{c})\text{ is equal to }(\mathrm{a},\mathrm{b},\mathrm{c}\in \mathrm{R})$

Solution:

Since, the second equation has imaginary roots. 
$\begin{array}{r}\therefore \frac{2\mathrm{a}}{3}=\frac{-3\mathrm{b}}{-4}=\frac{4\mathrm{c}}{5}=\mathrm{k}\\ \Rightarrow \mathrm{a}=\frac{3\mathrm{k}}{2},\mathrm{b}=\frac{4\mathrm{k}}{3},\mathrm{c}=\frac{5\mathrm{k}}{4}\\ \therefore \frac{\mathrm{a}+\mathrm{b}}{\mathrm{b}+\mathrm{c}}=\frac{\frac{3\mathrm{k}}{2}+\frac{4\mathrm{k}}{3}}{\frac{4\mathrm{k}}{3}+\frac{5\mathrm{k}}{4}}=\frac{34}{31}\end{array}$