Question

Let $\alpha ,\beta$ are the roots of the equation $2x^2-3x-7=0$, then the quadratic equation whose roots are $\frac{\alpha }{\beta }$ and $\frac{\beta }{\alpha }$ is

• A. $14x^2-37x+14=0$
• B. $14x^2-37x-14=0$
• C. $14x^2+37x-14=0$
• D. $14x^2+37x+14=0$

Answer 1

The correct option is D $14x^2+37x+14=0$

$2x^2-3x-7=0$
From the given equation,
$\alpha +\beta =\frac{3}{2},\alpha \beta =-\frac{7}{2}$
When the roots are,
$\frac{\alpha }{\beta },\frac{\beta }{\alpha }$
Sum of roots
$\frac{\alpha }{\beta }+\frac{\beta }{\alpha }=\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }=\frac{(\alpha +\beta {)}^2-2\alpha \beta }{\alpha \beta }=\frac{\frac{9}{4}+7}{-\frac{7}{2}}=-\frac{37}{14}$
Product of roots
$\frac{\alpha }{\beta }\times \frac{\beta }{\alpha }=1$
Hence, required equation is
$x^2+\frac{37}{14}x+1=0\therefore 14x^2+37x+14=0$