JEE MAIN - Mathematics (2020 - 5th September Evening Slot - No. 10)
If $$\alpha $$ and $$\beta $$ are the roots of the equation,
7x2 – 3x – 2 = 0, then the value of
$${\alpha \over {1 - {\alpha ^2}}} + {\beta \over {1 - {\beta ^2}}}$$ is equal to :
7x2 – 3x – 2 = 0, then the value of
$${\alpha \over {1 - {\alpha ^2}}} + {\beta \over {1 - {\beta ^2}}}$$ is equal to :
$${1 \over {24}}$$
$${{27} \over {32}}$$
$${{27} \over {16}}$$
$${3 \over 8}$$
Explanation
Given, 7x2 – 3x – 2 = 0
$$ \therefore $$ $$\alpha $$ + $$\beta $$ = $${3 \over 7}$$
$$\alpha $$$$\beta $$ = - $${2 \over 7}$$
$${\alpha \over {1 - {\alpha ^2}}} + {\beta \over {1 - {\beta ^2}}}$$
= $${{\alpha + \beta - \alpha \beta \left( {\alpha + \beta } \right)} \over {1 - {\alpha ^2} - {\beta ^2} + {\alpha ^2}{\beta ^2}}}$$
= $${{{3 \over 7} + {2 \over 7}\left( {{3 \over 7}} \right)} \over {1 - {{\left( {\alpha + \beta } \right)}^2} + 2\alpha \beta + {{\left( { - {2 \over 7}} \right)}^2}}}$$
= $${{{3 \over 7} + {2 \over 7}\left( {{3 \over 7}} \right)} \over {1 - {{\left( {{3 \over 7}} \right)}^2} + 2\left( { - {2 \over 7}} \right) + {{\left( { - {2 \over 7}} \right)}^2}}}$$
= $${{27} \over {16}}$$
$$ \therefore $$ $$\alpha $$ + $$\beta $$ = $${3 \over 7}$$
$$\alpha $$$$\beta $$ = - $${2 \over 7}$$
$${\alpha \over {1 - {\alpha ^2}}} + {\beta \over {1 - {\beta ^2}}}$$
= $${{\alpha + \beta - \alpha \beta \left( {\alpha + \beta } \right)} \over {1 - {\alpha ^2} - {\beta ^2} + {\alpha ^2}{\beta ^2}}}$$
= $${{{3 \over 7} + {2 \over 7}\left( {{3 \over 7}} \right)} \over {1 - {{\left( {\alpha + \beta } \right)}^2} + 2\alpha \beta + {{\left( { - {2 \over 7}} \right)}^2}}}$$
= $${{{3 \over 7} + {2 \over 7}\left( {{3 \over 7}} \right)} \over {1 - {{\left( {{3 \over 7}} \right)}^2} + 2\left( { - {2 \over 7}} \right) + {{\left( { - {2 \over 7}} \right)}^2}}}$$
= $${{27} \over {16}}$$
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