First slide

Theory of equations

Question

If $α and β$ are the roots of the equation ${ax}^{2} + bx + c = 0$ , then the equation whose roots are $α + \frac{1}{β} and β + \frac{1}{α},$ is

Moderate

A

${acx}^{2} + (a + c) bx + (a + c)^{2} = 0$
B

${abx}^{2} + (a + c) bx + (a + c)^{2} = 0$
C

${acx}^{2} + (a + b) cx + (a + c)^{2} = 0$
D

None of the above

Solution

$Since, α and β are the roots of {ax}^{2} + bx + c = 0 . \Rightarrow α + β = - \frac{b}{a} and αβ = \frac{c}{a} . . . . . . . . . . . . . . . . . . . (i)$
If roots are $α + \frac{1}{β}, β + \frac{1}{α}$ then,
Sum of roots $= (α + \frac{1}{β}) + (β + \frac{1}{α}) = (α + β) + \frac{α + β}{αβ}$
$= \frac{- b}{ac} (a + c) . . . . . . [from Eq . (i)$
and product of roots $= (α + \frac{1}{β}) (β + \frac{1}{α})$
$\begin{aligned} = αβ + 1 + 1 + \frac{1}{αβ} = 2 + \frac{c}{a} + \frac{a}{c} \end{aligned}$ [from Eq. (i)]
$= \frac{2 ac + c^{2} + a^{2}}{ac} = \frac{(a + c)^{2}}{ac}$
Hence, required equation is given by
$\begin{array}{l} x^{2} - (sum of roots) x + (product of roots) = 0 \\ \Rightarrow x^{2} + \frac{b}{ac} (a + c) x + \frac{(a + c)^{2}}{ac} = 0 \\ \Rightarrow {acx}^{2} + (a + c) bx + (a + c)^{2} = 0 \end{array}$

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