First slide

Theory of equations

Question

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

Moderate

A

$\frac{1}{2}$
B

$\frac{3}{35}$
C

$\frac{34}{31}$
D

$\frac{29}{31}$

Solution

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

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