Question

If $\alpha \ne \beta$ but ${\alpha }^2=5\alpha -3$ and ${\beta }^2=5\beta -3$ then the equation whose roots are $\alpha /\beta$ and $\beta /\alpha$ is

• A. $3x^2-25x+3=0$
• B. $x^2+5x-3=0$
• C. $x^2-5x+3=0$
• D. $3x^2-19x+3=0$

Answer 1

The correct option is D $3x^2-19x+3=0$

Given ${\alpha }^2=5\alpha -3,{\beta }^2=5\beta -3$

$⟹\alpha ,\beta$ are the roots of $x^2-5x+3=0$

$⟹\alpha +\beta =5,\alpha \beta =3$

$\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }=\frac{5^2-6}{3}=\frac{19}{3}$

The equation with $\frac{\alpha }{\beta },\frac{\beta }{\alpha }$ roots is

$x^2-\frac{19}{3}x+1=0⟹3x^2-19x+3=0$