First slide

Theory of equations

Question

If $α$ and $β$ are the roots of the equation, 7x² -3x - 2 = 0,then the value of $\frac{α}{1 - α^{2}} + \frac{β}{1 - β^{2}}$ is equal to

Moderate

A

$\frac{27}{32}$
B

$\frac{1}{24}$
C

$\frac{3}{8}$
D

$\frac{27}{16}$

Solution

Given quadratic equation 7x² -3x- 2 =0 has roots $α$ and $β$ .
$So, α + β = \frac{3}{7} and αβ = - \frac{2}{7} .$
Now,
$\frac{α}{1 - α^{2}} + \frac{β}{1 - β^{2}} = \frac{α - {αβ}^{2} + β - {βα}^{2}}{1 - (α^{2} + β^{2}) + α^{2} β^{2}}$
$\begin{array}{l} = \frac{(α + β) - αβ (α + β)}{1 - [(α + β)^{2} - 2 αβ] + (αβ)^{2}} \\ = \frac{\frac{3}{7} + \frac{2}{7} (\frac{3}{7})}{1 - [{(\frac{3}{7})}^{2} + \frac{4}{7}] + \frac{4}{49}} \\ = \frac{21 + 6}{49 - 37 + 4} = \frac{27}{16} \end{array}$

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