JEE Main 2020 · 5 September · Evening Shift · Mathematics · Quadratic Equation
D. 2716\dfrac{27}{16}1627
α+β=37;αβ=−27\alpha + \beta = \dfrac{3}{7}; \alpha\beta = - \dfrac{2}{7}α+β=73;αβ=−72
Now
α1−α2+β1−β2=(α+β)−αβ(α+β)(1−α2)(1−β2)=(α+β)−αβ(α+β)1+(αβ)2−(α2+β2)\dfrac{\alpha}{1 - {\alpha}^{2}} + \dfrac{\beta}{1 - {\beta}^{2}} = \dfrac{( \alpha + \beta ) - \alpha\beta ( \alpha + \beta )}{\left(1 - {\alpha}^{2}\right)\left(1 - {\beta}^{2}\right)} = \dfrac{( \alpha + \beta ) - \alpha\beta ( \alpha + \beta )}{1 + ( \alpha\beta{)}^{2} - \left({\alpha}^{2} + {\beta}^{2}\right)}1−α2α+1−β2β=(1−α2)(1−β2)(α+β)−αβ(α+β)=1+(αβ)2−(α2+β2)(α+β)−αβ(α+β)
⇒(α+β)−αβ(α+β)1+(αβ)2−(α+β)2+2αβ=37+27(37)1+(27)2−(37)2−2(27)=2716⇒ \dfrac{( \alpha + \beta ) - \alpha\beta ( \alpha + \beta )}{1 + ( \alpha\beta{)}^{2} - ( \alpha + \beta{)}^{2} + 2\alpha\beta} = \dfrac{\dfrac{3}{7} + \dfrac{2}{7}\left(\dfrac{3}{7}\right)}{1 + {\left(\dfrac{2}{7}\right)}^{2} - {\left(\dfrac{3}{7}\right)}^{2} - 2\left(\dfrac{2}{7}\right)} = \dfrac{27}{16}⇒1+(αβ)2−(α+β)2+2αβ(α+β)−αβ(α+β)=1+(72)2−(73)2−2(72)73+72(73)=1627
