First slide

Theory of equations

Question

If $α, β$ 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 these

Solution

Here $α + β = - \frac{b}{a} and αβ = \frac{c}{a}$
If roots are $α + \frac{1}{β}, β + \frac{1}{α},$ then sum of roots are
$= (α + \frac{1}{β}) + (β + \frac{1}{α}) = (α + β) + \frac{α + β}{αβ} = - \frac{b}{ac} (a + c)$
and product $= (α + \frac{1}{β}) (β + \frac{1}{α})$
$= αβ + 1 + 1 + \frac{1}{αβ} = 2 + \frac{c}{a} + \frac{a}{c} = \frac{2 ac + c^{2} + a^{2}}{ac} = \frac{{(a + c)}^{2}}{ac}$
Hence required equation is given by
$x^{2} + \frac{b}{a c} (a + c) x + \frac{{(a + c)}^{2}}{a c} = 0$
$\Rightarrow {acx}^{2} + (a + c) bx + {(a + c)}^{2} = 0$

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