Question

If $\alpha \ne \beta ,{\alpha }^2=5\alpha -3,{\beta }^2=5\beta -3$ then the equation whose roots are $\frac{\alpha }{\beta }$ & $\frac{\beta }{\alpha }$ is

• A. $x^2+5x-3=0$
• B. $3x^2+12x+3=0$
• C. $3x^2-19x+3=0$
• D. None of these

Answer 1

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

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

$\alpha +\beta =5,\alpha \beta =3$

Eqn whose roots are $\alpha /\beta$ & $\beta /\alpha$ is

$x^2-(\frac{\alpha }{\beta }+\frac{\beta }{\alpha })x+(\frac{\alpha }{\beta }\times \frac{\beta }{\alpha })=0$

$\Rightarrow x^2-\left(\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }\right)x+1=0$

$\Rightarrow x^2-\left(\frac{{(\alpha +\beta )}^2-2\alpha \beta }{\alpha \beta }\right)x+1=0\Rightarrow x^2-\frac{19}{3}x+1=0$

$\Rightarrow {3x}^2-19x+3=0$ [C]