Question

If $\alpha$ and $\beta$ are the roots of the equation $6y^2-7y+2$, Form a quadratic equation whose roots are $\frac{1}{\alpha }$ and $\frac{1}{\beta }$.

Answer 1

Sum of the roots$=\alpha +\beta =\frac{7}{6}$

Product of the roots$=\alpha \beta =\frac{2}{6}$

To form a quadratic equation of roots $\frac{1}{\alpha },\frac{1}{\beta }$

$\therefore$Sum of the roots$=\frac{1}{\alpha }+\frac{1}{\beta }$

$=\frac{\alpha +\beta }{\alpha \beta }=\frac{\frac{7}{6}}{\frac{2}{6}}=\frac{7}{2}$

Product of the roots$=\frac{1}{\alpha }\times \frac{1}{\beta }$

$=\frac{1}{\alpha \beta }=\frac{1}{\frac{2}{6}}=\frac{6}{2}=3$

$\therefore$ Required equation is $x^2-(\frac{1}{\alpha }+\frac{1}{\beta })x+\frac{1}{\alpha \beta }=0$

$\Rightarrow x^2-\left(\frac{7}{2}\right)x+3=0$

$\Rightarrow 2x^2-7x+6=0$