Question

if x is equal to under root a + under root b upon under root a minus under root b and Y is equal to under root a minus under root b upon under root a + under root b then find the value of x square + xy + Y square

Answer 1

simplify.....to get the solution

Answer 2

$x = \frac{(\sqrt{a} + \sqrt{b})}{(\sqrt{a} - \sqrt{b})} \\and \: y = \frac{(\sqrt{a} - \sqrt{b})}{(\sqrt{a} + \sqrt{b})}$

$i ) x + y \\= </p><p>\frac{(\sqrt{a} + \sqrt{b})}{(\sqrt{a} - \sqrt{b})} + \frac{(\sqrt{a} - \sqrt{b})}{(\sqrt{a} + \sqrt{b})}$

$= \frac{(\sqrt{a} + \sqrt{b})^{2} + (\sqrt{a} - \sqrt{b})^{2} } { (\sqrt{a} + \sqrt{b})(\sqrt{a} - \sqrt{b})}\\= \frac{ 2[ \sqrt{a}^{2} + \sqrt{b}^{2} ] }{ (\sqrt{a}^{2} - \sqrt{b}^{2} )}$

$= \frac{2( a+b)}{(a-b)} \: --(1)$

$ii )xy = \frac{(\sqrt{a} + \sqrt{b})}{(\sqrt{a} - \sqrt{b})} \times \frac{(\sqrt{a} - \sqrt{b})}{(\sqrt{a} + \sqrt{b})} \\= 1 \: --(2)$

$iii ) Now, Value \: of \: x^{2} + xy + y^{2} \\= x^{2} + 2xy + y^{2} - xy \\= (x+y)^{2} - xy \\= \Big ( \frac{2( a+b)}{(a-b)}\Big)^{2} - 1$

$= \frac{4(a+b)^{2} - (a-b)^{2}}{(a-b)^{2} } \\= \frac{4(a^{2} + b^{2} + 2ab) - a^{2}-b^{2} + 2ab}{(a-b)^{2} }\\= \frac{ 3a^{2} + 3b^{2} + 10ab}{(a-b)^{2} }$

Therefore.,

$\red { Value \: of \: x^{2} + xy + y^{2}} \\\green {= </p><p>\frac{ 3a^{2} + 3b^{2} + 10ab}{(a-b)^{2} }}$

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