First slide

Theory of equations

Question

$If a, b, c, d are the roots of the equation x^{4} - 2 π x - \sqrt{2019} = 0, then the product of$ $\frac{a + b + c}{a b c}, \frac{b + c + d}{b c d}, \frac{c + d + a}{c d a}, \frac{d + a + b}{a b d} is equal to$

Moderate

A

$\frac{1}{\sqrt{2019}}$
B

$\frac{1}{2019}$
C

$\sqrt{2019}$
D

$- \sqrt{2019}$

Solution

$Given equation is x^{4} - 2 π x - \sqrt{2019} = 0 - - - - - - (1)$
$Since a, b, c, d are the roots of equation (1), then$
$Sum of the roots S_{1} = a + b + c + d = 0 - - - - - - - (2)$
$Product of the roots S_{4} = a b c d = \sqrt{2019} - - - - - - (3)$
Now from equation (1)
, $a + b + c = - d$
$b + c + d = - a$
$c + d + a = - b$
$d + a + b = - c$
[SolutionStep4]:
$∴ Required Product = (\frac{a + b + c}{a b c}) (\frac{b + c + d}{b c d}) (\frac{c + d + a}{c d a}) (\frac{d + a + b}{a d b})$
$\begin{array}{l} = (\frac{- d}{a b c}) (\frac{- a}{b c d}) (\frac{- b}{c d a}) (\frac{- c}{a b d}) \\ = \frac{1}{{(a b c d)}^{2}} \\ = \frac{1}{2019} \end{array}$ $(∵ from equation (3))$

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