Question

If $x = \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}},y = \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}},$ then $3x^{2} + 4xy - 3y^{2} =$

• A. $\frac{1}{3}\lbrack 56\sqrt{10} - 12\rbrack$
• B. $\frac{1}{3}\lbrack 56\sqrt{10} + 12\rbrack$
• C. $\frac{1}{3}\lbrack 56 + 12\sqrt{10}\rbrack$
• D. None of these

Answer 1

Answer: (B)

$\frac{1}{3}\lbrack 56\sqrt{10} + 12\rbrack$

Answer 2

$y = \frac{1}{x}$ $\Rightarrow$ $xy = 1$

$\therefore$ $3x^{2} + 4xy - 3y^{2} = 3(x - y)(x + y + 4)$

$= 3.\left( \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} - \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} \right)\left( \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} - \sqrt{2}} + \frac{\sqrt{5} - \sqrt{2}}{\sqrt{5} + \sqrt{2}} \right) + 4$

$= \frac{3\lbrack(\sqrt{5} + \sqrt{2})^{2} - (\sqrt{5} - \sqrt{2})^{2}\rbrack}{(5 - 2)(5 - 2)}\lbrack(\sqrt{5} + \sqrt{2})^{2} + (\sqrt{5} - \sqrt{2})^{2}\rbrack + 4$

$= \frac{1}{3}.4\sqrt{10}.2(5 + 2) + 4 = \frac{56}{3}\sqrt{10} + 4 = \frac{1}{3}(56\sqrt{10} + 12)$.