If a = $\frac{\sqrt5+1}{\sqrt5-1}$ and b = $\frac{\sqrt5-1}{\sqrt5+1}$, then the value of $\frac{a^2 + ab + b^2}{a^2 - ab + b^2}$ is:

Question

Accepted Answer

To solve the problem and justify why Option 4 is correct, we need to evaluate the expression $\frac{a^2 + ab + b^2}{a^2 - ab + b^2}$ given the values of $a$ and $b$.

First, let's simplify $a$ and $b$:

1. $a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}$

To simplify $a$, multiply the numerator and the denominator by the conjugate of the denominator:

$$
   a = \frac{(\sqrt{5} + 1)(\sqrt{5} + 1)}{(\sqrt{5} - 1)(\sqrt{5} + 1)} = \frac{(\sqrt{5})^2 + 2\sqrt{5} + 1}{5 - 1} = \frac{5 + 2\sqrt{5} + 1}{4} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2}
   $$

2. $b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1}$

Similarly, simplify $b$ by multiplying by the conjugate:

$$
   b = \frac{(\sqrt{5} - 1)(\sqrt{5} - 1)}{(\sqrt{5} + 1)(\sqrt{5} - 1)} = \frac{(\sqrt{5})^2 - 2\sqrt{5} + 1}{5 - 1} = \frac{5 - 2\sqrt{5} + 1}{4} = \frac{6 - 2\sqrt{5}}{4} = \frac{3 - \sqrt{5}}{2}
   $$

Now, calculate $a^2$, $b^2$, and $ab$:

- $a^2 = \left(\frac{3 + \sqrt{5}}{2}\right)^2 = \frac{(3 + \sqrt{5})^2}{4} = \frac{9 + 6\sqrt{5} + 5}{4} = \frac{14 + 6\sqrt{5}}{4} = \frac{7 + 3\sqrt{5}}{2}$

- $b^2 = \left(\frac{3 - \sqrt{5}}{2}\right)^2 = \frac{(3 - \sqrt{5})^2}{4} = \frac{9 - 6\sqrt{5} + 5}{4} = \frac{14 - 6\sqrt{5}}{4} = \frac{7 - 3\sqrt{5}}{2}$

- $ab = \left(\frac{3 + \sqrt{5}}{2}\right)\left(\frac{3 - \sqrt{5}}{2}\right) = \frac{(3 + \sqrt{5})(3 - \sqrt{5})}{4} = \frac{9 - (\sqrt{5})^2}{4} = \frac{9 - 5}{4} = 1$

Now substitute these into the expression $\frac{a^2 + ab + b^2}{a^2 - ab + b^2}$:

- Numerator: $a^2 + ab + b^2 = \frac{7 + 3\sqrt{5}}{2} + 1 + \frac{7 - 3\sqrt{5}}{2} = \frac{7 + 3\sqrt{5} + 7 - 3\sqrt{5}}{2} + 1 = \frac{14}{2} + 1 = 7 + 1 = 8$

- Denominator: $a^2 - ab + b^2 = \frac{7 + 3\sqrt{5}}{2} - 1 + \frac{7 - 3\sqrt{5}}{2} = \frac{7 + 3\sqrt{5} + 7 - 3\sqrt{5}}{2} - 1 = \frac{14}{2} - 1 = 7 - 1 = 6$

Thus, the expression simplifies to:

$$
\frac{a^2 + ab + b^2}{a^2 - ab + b^2} = \frac{8}{6} = \frac{4}{3}
$$

Therefore, the correct answer is Option 4: $\frac{4}{3}$.

Explanation of Incorrect Options:

1. Option 1: $\frac{1}{3}$ is incorrect because the calculated value of the expression is $\frac{4}{3}$, not $\frac{1}{3}$.

2. Option 2: 1 is incorrect because the expression simplifies to $\frac{4}{3}$, not 1.

3. Option 3: $\frac{2}{3}$ is incorrect because the expression simplifies to $\frac{4}{3}$, not $\frac{2}{3}$.

Thus, Option 4 is the only correct choice.