A) \[{{30}^{o}}\]
B) \[{{45}^{o}}\]
C) \[{{60}^{o}}\]
D) \[{{75}^{o}}\]
Correct Answer: D
Solution :
Let the required line through the point (1,2) be inclined at an angle \[\theta \] to the axis of x. Then its equation is \[\frac{x-1}{\cos \theta }=\frac{y-2}{\sin \theta }=r\] .....(i) where r is distance of any point (x, y) on the line from the point (1, 2). The coordinates of any point on the line (i) are \[(1+r\cos \theta ,\text{ }2+r\sin \theta )\]. If this point is at a distance \[\frac{\sqrt{6}}{3}\] form (1, 2), then \[r=\frac{\sqrt{6}}{3}.\] Therefore, the point is \[\left( 1+\frac{\sqrt{6}}{3}\cos \theta ,\text{ }2+\frac{\sqrt{6}}{3}\sin \theta \right)\]. But this point lies on the line \[x+y=4\]. Þ \[\frac{\sqrt{6}}{3}(\cos \theta +\sin \theta )=1\] or \[\sin \theta +\cos \theta =\frac{3}{\sqrt{6}}\] Þ \[\frac{1}{\sqrt{2}}\sin \theta +\frac{1}{\sqrt{2}}\cos \theta =\frac{\sqrt{3}}{2}\], {Dividing both sides by\[\sqrt{2}\]} Þ \[\sin (\theta +{{45}^{o}})=\sin {{60}^{o}}\]or sin \[{{120}^{o}}\] Þ \[\theta ={{15}^{o}}\]or \[{{75}^{o}}\].
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