Question

If $\tan \left(\frac{\pi }{8}\right)$ is a root of the equation $x^2+ax+b=0,$ where $a$ and $b$ are rational numbers, then the value of $a^2+b^2$ is

• A. $5$
• B. $7$
• C. $8$
• D. $9$

Answer 1

The correct option is A $5$

We know that $\tan \left(\frac{\pi }{8}\right)=\sqrt{2}-1$
Since the coefficients of $x^2+ax+b=0$ are rational,
$\therefore$ other root is $-\sqrt{2}-1.$
Now, sum of the roots, $-a=(\sqrt{2}-1)+(-\sqrt{2}-1)$
$\Rightarrow a=2$
Product of the roots, $b=(\sqrt{2}-1)(-\sqrt{2}-1)$
$\Rightarrow b=-1$
$\therefore a^2+b^2=5$