Question

Sol. [Key Concept: Distance form on $x + y = 4$ gives $\sin\ \theta = \frac{12}{}$; the line through $(1,2)$ makes $\frac{\pi}{4} \pm \frac{\pi}{6}$.]

$$sin\theta = \frac{\frac{1}{\sqrt{2}}}{\frac{\sqrt{6}}{3}} = \frac{1}{2} \Rightarrow \theta = \frac{\pi}{6}$$

$\therefore\ $ Angle made by $L$ with positive $x$-axis can be

$= \frac{\pi}{4} - \frac{\pi}{6}$ and $\frac{\pi}{4} + \frac{\pi}{6}$

$= \frac{\pi}{12}$ and $\frac{5\pi}{12}$

✔ (a) $\frac{\pi}{12} = \frac{\pi}{4} - \frac{\pi}{6}$.

✘ (b) $\frac{\pi}{6}$ is not obtained.

✘ (c) $\frac{\pi}{3}$ is not obtained.

✔ (d) $\frac{5\pi}{12} = \frac{\pi}{4} + \frac{\pi}{6}$.