Question

If $\tan {22}^0$ and $\tan {23}^0$ are roots of $x^2+ax+b=0$, then

• A. $a+b+1=0$
• B. $a-b+1=0$
• C. $b-a+1=0$
• D. $a+b=1$

Answer 1

The correct option is B $a-b+1=0$

Given:

$\alpha =\tan {22}^{\circ },\beta =\tan {23}^{\circ }$

where $\alpha$ and $\beta$ are roots of the equation $x^2+ax+b=0.$

Therefore,

$\alpha +\beta =\tan {22}^{\circ }+\tan {23}^{\circ }=-a$ $...\left(1\right)$

$\alpha \beta =\tan {22}^{\circ }\times \tan {23}^{\circ }=b$ $...\left(2\right)$

We know

$\tan {45}^{\circ }=\tan ({22}^{\circ }+{23}^{\circ })=\frac{\tan {22}^{\circ }+\tan {23}^{\circ }}{1-\tan {22}^{\circ }.\tan {23}^{\circ }}$

From $\left(1\right)$ and $\left(2\right),$

$\frac{-a}{1-b}=\tan {45}^{\circ }=1$

$\Rightarrow a-b+1=0$

Hence, option $B.$