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$\alpha$ and $\beta$ are zeroes of a quadratic polynomial $px^2+qx+1$. Form a quadratic polynomial whose zeroes are $\frac{2}{\alpha}$ and $\frac{2}{\beta}$.
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$\alpha + \beta = -\frac{q}{p}, \alpha\beta = \frac{1}{p}$
Sum of zeroes of the required polynomial = $\frac{2}{\alpha} + \frac{2}{\beta} = 2\frac{(\beta+\alpha)}{\alpha\beta} = -2q$
Product of zeroes of the required polynomial = $\frac{2}{\alpha} \times \frac{2}{\beta} = \frac{4}{\alpha\beta} = 4p$
$\therefore$ required polynomial is $x^2 + 2qx + 4p$
Sum of zeroes of the required polynomial = $\frac{2}{\alpha} + \frac{2}{\beta} = 2\frac{(\beta+\alpha)}{\alpha\beta} = -2q$
Product of zeroes of the required polynomial = $\frac{2}{\alpha} \times \frac{2}{\beta} = \frac{4}{\alpha\beta} = 4p$
$\therefore$ required polynomial is $x^2 + 2qx + 4p$