A) \[1+\sqrt{3}\]
B) \[2-\sqrt{3}\]
C) \[3+\sqrt{5}\]
D) \[2-\sqrt{5}\]
Correct Answer: D
Solution :
Given, direction ratios of given line are \[(\lambda +1,\,1-\lambda ,\,2).\] Then, \[\sqrt{{{(\lambda +1)}^{2}}+{{(1-\lambda )}^{2}}+{{(2)}^{2}}}\] \[=\sqrt{{{\lambda }^{2}}+1+2\lambda +1-2\lambda +{{\lambda }^{2}}+4}=\sqrt{2{{\lambda }^{2}}+6}\] \[\therefore \] Direction cosines of line are \[\frac{\lambda +1}{\sqrt{2{{\lambda }^{2}}+6}},\,\,\frac{1-\lambda }{\sqrt{2{{\lambda }^{2}}+6}},\,\frac{2}{\sqrt{2{{\lambda }^{2}}+6}}\] Also given, line makes an angle of \[{{60}^{o}}\] with Y axis. \[\therefore \] \[\cos {{60}^{o}}=\frac{1-\lambda }{\sqrt{2{{\lambda }^{2}}+6}}\] \[\Rightarrow \] \[\frac{1}{2}=\frac{1-\lambda }{\sqrt{2{{\lambda }^{2}}+6}}\] \[\Rightarrow \] \[\sqrt{2{{\lambda }^{2}}+6}=2(1-\lambda )\] \[\Rightarrow \] \[2{{\lambda }^{2}}+6=4{{(1-\lambda )}^{2}}\] [on squaring both sides] \[\Rightarrow \] \[2{{\lambda }^{2}}+6\lambda =4+4{{\lambda }^{2}}-8\lambda \] \[\Rightarrow \] \[2{{\lambda }^{2}}-8-2=0\] \[\therefore \]\[\lambda =\frac{-(-4)\pm \sqrt{{{(-4)}^{2}}-4(-1)}}{2}=\frac{4\pm \sqrt{20}}{2}\] \[=\frac{4\pm 2\sqrt{5}}{2}=2\pm \sqrt{5}\]
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