Two lines L_1 and L_2 passing through the poi | vedclass

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(C) Let the line passing through $P(1, 2)$ make an angle $	heta$ with the positive $X$-axis. The parametric form of the line is given by $x = 1 + r \

Vedclass · Accepted Answer

$\frac{\pi}{12}, \frac{5 \pi}{12}$

Vedclass · Answer

(C) Let the line passing through $P(1, 2)$ make an angle $	heta$ with the positive $X$-axis. The parametric form of the line is given by $x = 1 + r \cos 	heta$ and $y = 2 + r \sin 	heta$,where $r = \frac{\sqrt{6}}{3}$.Since the point $(x, y)$ lies on the line $x + y = 4$,we substitute the parametric coordinates:$(1 + r \cos 	heta) + (2 + r \sin 	heta) = 4$$3 + r(\cos 	heta + \sin 	heta) = 4$$r(\cos 	heta + \sin 	heta) = 1$Substituting $r = \frac{\sqrt{6}}{3}$:$\frac{\sqrt{6}}{3}(\cos 	heta + \sin 	heta) = 1$$\cos 	heta + \sin 	heta = \frac{3}{\sqrt{6}} = \frac{\sqrt{6}}{2}$Squaring both sides:$(\cos 	heta + \sin 	heta)^2 = \frac{6}{4} = \frac{3}{2}$$1 + \sin 2	heta = \frac{3}{2}$$\sin 2	heta = \frac{1}{2}$Thus,$2	heta = \frac{\pi}{6}$ or $2	heta = \frac{5\pi}{6}$.Therefore,$	heta = \frac{\pi}{12}$ or $	heta = \frac{5\pi}{12}$.

Two lines $L_1$ and $L_2$ passing through the point $P(1, 2)$ cut the line $x+y=4$ at a distance of $\frac{\sqrt{6}}{3}$ units from $P$. Then the angles made by $L_1$ and $L_2$ with the positive $X$-axis are

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