If - but -2-5 -3- -2-5 -3- then the equation whose roots are - and - is

First we will find the sum and product of roots of required eqution. Here,S=αβ+β/α=α^2+β^2αβ =5 α 3+5 β 3αβ[[ ∵α^2=5 α 3; β^2=5 β 3 ]] S=5(α+β) 6αβ and nbsp; ...

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Standard XII

Mathematics

Relations Between Roots and Coefficients

If α≠β, but α...

Question

If $α \neq β$ , but $α^{2} = 5 α - 3, β^{2} = 5 β - 3$ , then the equation whose roots are $\frac{α}{β}$ and $\frac{β}{α}$ is

3 x^{2} + 12 x + 3 = 0

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3 x^{2} - 19 x + 3 = 0

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None of these

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x^{2} - 5 x - 3 = 0

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Solution

The correct option is B $3 x^{2} - 19 x + 3 = 0$
First we will find the sum and product of roots of required eqution.
Here, $S = \frac{α}{β} + \frac{β}{α} = \frac{α^{2} + β^{2}}{α β} = \frac{5 α - 3 + 5 β - 3}{α β} [\begin{matrix} ∵ α^{2} = 5 α - 3 β^{2} = 5 β - 3 \end{matrix}]$
$S = \frac{5 (α + β) - 6}{α β} a n d P = \frac{α}{β} \cdot \frac{β}{α} = 1 \Rightarrow P = 1 . α, β are roots of x^{2} - 5 x + 3 = 0$ .
Therefore $α + β = 5, α β = 3 S = \frac{5 (5) - 6}{3} = \frac{19}{3}$
As we know a quadratic equation with given roots is given by
$x^{2} - (Sum of roots) x + Product of roots = 0$
$\Rightarrow x^{2} - \frac{19}{3} x + 1 = 0 \Rightarrow 3 x^{2} - 19 x + 3 = 0$

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If α≠β, but α2=5α−3,β2=5β−3, then the equation whose roots are αβ and βα is

If $α \neq β$ , but $α^{2} = 5 α - 3, β^{2} = 5 β - 3$ , then the equation whose roots are $\frac{α}{β}$ and $\frac{β}{α}$ is