Question

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

Answer: (C)

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

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

So we can say roots of $x^2-5x+3=0$ are $\alpha ,\beta$
$\alpha +\beta =5$
$\alpha \beta =3$

Now, for equation whose roots are$\frac{\alpha }{\beta }and\frac{\beta }{\alpha }$

Sum of the roots $=\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{({\alpha }^2+{\beta }^2)}{\alpha \beta }$
$=\frac{(\alpha +\beta {)}^2-2\alpha \beta }{\alpha \beta }$
$=\frac{25-6}{3}=\frac{19}{3}$
Product of roots = $\frac{\alpha }{\beta }\times \frac{\beta }{\alpha }$
$=1$

So, the required equation is

$x^2-\left(sumofroots\right)x+productofroots=0$

$x^2-\frac{19}{3}x+1=0$

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