Question

if a = √5 + 1/√5 - 1 and b = √5-1/√5 + 1, the value of a² + ab + b²/a² - ab + b² is​

Answer 1

Hint.

Rationalizing $a=\dfrac{\sqrt{5} +1}{\sqrt{5} -1}$ and $b=\dfrac{\sqrt{5} -1}{\sqrt{5} +1}$ we obtain $a=\dfrac{3+\sqrt{5} }{2}$ and $b=\dfrac{3-\sqrt{5} }{2}$.

We observe that $a+b=3$ and $ab=1$.

Solution.

We have,

• $a+b=3$
• $ab=1$

Given, $\dfrac{a^2+ab+b^2}{a^2-ab+b^2}$

$=\dfrac{(a+b)^2-ab}{(a+b)^2-3ab}$

$=\dfrac{3^2-1}{3^2-3} =\dfrac{8}{6} =\boxed{\frac{4}{3} }$

Answer 2

Firstly,

Rationalizing $a=\dfrac{\sqrt{5} +1}{\sqrt{5} -1}$ and $b=\dfrac{\sqrt{5} -1}{\sqrt{5} +1}$ we obtain $a=\dfrac{3+\sqrt{5} }{2}$ and $b=\dfrac{3-\sqrt{5} }{2}$.

We observe that $a+b=3$ and $ab=1$.

Now,

We have,

$a+b=3$

$ab=1$

Given,

$\displaystyle\dfrac{a^2+ab+b^2}{a^2-ab+b^2}$

$=\dfrac{(a+b)^2-ab}{(a+b)^2-3ab}$

$=\dfrac{3^2-1}{3^2-3} =\dfrac{8}{6} ={\frac{4}{3} }$