Question

If $\alpha$ and $\beta$ are the roots of the equation $4x^2-5x+2=0$, find the equation whose roots are

$\frac{\alpha }{\beta }$ and $\frac{\beta }{\alpha }$.

• A. $8x^2+9x+8=0$
• B. $8x^2-9x+8=0$
• C. $8x^2-9x-8=0$
• D. $x^2-9x+8=0$

Answer 1

The correct option is B $8x^2-9x+8=0$

$4x^2-5x+2=0$

If $\alpha$ and $\beta$ are the roots of this equation,

then , sum of roots: $\alpha +\beta$ = $\frac{5}{4}$

Product of roots: $\alpha .\beta =\frac{2}{4}$
The equation which has roots as : $\frac{\alpha }{\beta }$ and $\frac{\beta }{\alpha }$

Sum of roots: $\frac{\alpha }{\beta }+\frac{\beta }{\alpha }$

= $\frac{{\alpha }^2+{\beta }^2}{\alpha \beta }$

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

= $\frac{{\left(\frac{5}{4}\right)}^2-2\frac{2}{4}}{\frac{2}{4}}$

= $\frac{9}{8}$

Product of roots: $\left(\frac{\alpha }{\beta }\right)\left(\frac{\beta }{\alpha }\right)=1$

Thus new equation is :$x^2-Sx+P=0$

$x^2-\frac{9}{8}+1=0$

$8x^2-9x+8=0$