If α≠β and α2=5α−3,β2=5β−3, then the equation αβ and βα as its roots, is

Given α2=5α−3⇒ α2−5α+3=0 and β2=5β−3⇒ β2−5β+3=0These two equations shows that α andβ are the roots of the equation x2−5x+3=0∴ α+β=5 and αβ=3Now, αβ+βα=α2+β2αβ=(α+β)2−2αβαβ=25−63=193and αβ⋅βα=1Thus, the equation having roots αβ and βα is given by x2−αβ+βαx+αβ⋅βα=0⇒ x2−193x+1=0⇒ 3x2−19x+3=0

If α≠β and α2=5α−3,β2=5β−3, then the equation αβ and βα as its roots, is

Given α2=5α−3⇒ α2−5α+3=0 and β2=5β−3⇒ β2−5β+3=0These two equations shows that α andβ are the roots of the equation x2−5x+3=0∴ α+β=5 and αβ=3Now, αβ+βα=α2+β2αβ=(α+β)2−2αβαβ=25−63=193and αβ⋅βα=1Thus, the equation having roots αβ and βα is given by x2−αβ+βαx+αβ⋅βα=0⇒ x2−193x+1=0⇒ 3x2−19x+3=0

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

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

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

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

$x^{2} - 16 x + 1 = 0$

answer is B.

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Detailed Solution

Given $α^{2} = 5 α - 3$

$\begin{array}{lrl} \Rightarrow α^{2} - 5 α + 3 = 0 \\ and β^{2} = 5 β - 3 \\ \Rightarrow β^{2} - 5 β + 3 = 0 \end{array}$

These two equations shows that $α$ and $β$ are the roots of the equation $x^{2} - 5 x + 3 = 0$

$∴ α + β = 5 and α β = 3$

Now, $\frac{α}{β} + \frac{β}{α} = \frac{α^{2} + β^{2}}{α β}$

$= \frac{(α + β)^{2} - 2 α β}{α β} = \frac{25 - 6}{3} = \frac{19}{3}$

and $\frac{α}{β} \cdot \frac{β}{α} = 1$

Thus, the equation having roots $\frac{α}{β} and \frac{β}{α}$ is given by

$\begin{array}{l} x^{2} - (\frac{α}{β} + \frac{β}{α}) x + \frac{α}{β} \cdot \frac{β}{α} = 0 \\ \Rightarrow x^{2} - \frac{19}{3} x + 1 = 0 \\ \Rightarrow 3 x^{2} - 19 x + 3 = 0 \end{array}$