A) \[{{x}^{2}}-\sqrt{3}(1-a)x+a=0\]
B) \[\sqrt{3}{{x}^{2}}-(1-a)x+a\sqrt{3}=0\]
C) \[{{x}^{2}}+\sqrt{3}(1+a)x-a=0\]
D) \[\sqrt{3}{{x}^{2}}+(1+a)x-a\sqrt{3}=0\]
Correct Answer: B
Solution :
\[\because \] \[\tan x.\tan y=a\] and \[\tan (x+y)=\tan \left( \frac{\pi }{6} \right)\] \[\Rightarrow \] \[\frac{\tan x+\tan y}{1-\tan x.\tan \,y}=\frac{1}{\sqrt{3}}\] \[\Rightarrow \] \[\tan x+\tan y=\frac{1}{\sqrt{3}}(1-a)\] Equation whose roots are \[\tan x\]and \[\tan y\]is \[{{x}^{2}}-\frac{(1-a)}{\sqrt{3}}.x+a=0\] \[\Rightarrow \] \[\sqrt{3}{{x}^{2}}-(1-a)x+a\sqrt{3}=0\]
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