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If $\alpha, \beta$ are the roots of the equation $\ell \mathrm{x}^{2}-\mathrm{mx}+\mathrm{m}=0$, $\ell \neq \mathrm{m}, \ell \neq 0$, then which one of the following statements is
correct?
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correct?
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Verified Answer
The correct answer is:
$\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}-\sqrt{\frac{\mathrm{m}}{\ell}}=0$
As given : $\alpha$ and $\beta$ are the roots of the quadratio equation $\ell \mathrm{x}^{2}-\mathrm{mx}+\mathrm{m}=0$
So, sum of roots,
$\alpha+\beta=\frac{\mathrm{m}}{\ell}$ and product of roots, $\alpha \beta=\frac{\mathrm{m}}{\ell}$
$\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}=\frac{\alpha+\beta}{\sqrt{\alpha \beta}}=\frac{\mathrm{m} / \ell}{\sqrt{\mathrm{m} / \ell}}$
$\Rightarrow \sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}=\sqrt{\frac{\mathrm{m}}{\ell}}=0 \Rightarrow \sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}-\sqrt{\frac{\mathrm{m}}{\ell}}=0$
So, sum of roots,
$\alpha+\beta=\frac{\mathrm{m}}{\ell}$ and product of roots, $\alpha \beta=\frac{\mathrm{m}}{\ell}$
$\sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}=\frac{\alpha+\beta}{\sqrt{\alpha \beta}}=\frac{\mathrm{m} / \ell}{\sqrt{\mathrm{m} / \ell}}$
$\Rightarrow \sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}=\sqrt{\frac{\mathrm{m}}{\ell}}=0 \Rightarrow \sqrt{\frac{\alpha}{\beta}}+\sqrt{\frac{\beta}{\alpha}}-\sqrt{\frac{\mathrm{m}}{\ell}}=0$
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