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If the line x \cos \theta+y \sin \theta=2 is the equation of a transverse common tangent to the circles \mathrm{x^{2}+y^{2}=4} and \mathrm{x^{2}+y^{2}-6 \sqrt{3} x-6 y+20=0}, then the value of \theta is:

Option: 1

5 \pi / 6


Option: 2

2 \pi / 3


Option: 3

\pi / 3


Option: 4

\pi / 6


Answers (1)

best_answer

\mathrm{C}_{1} \mathrm{C}_{2}=\mathrm{r}_{1}+\mathrm{r}_{2}
\mathrm{\mathrm{C}_{1}=(0,0) ; \mathrm{C}_{2}=(3 \sqrt{3}, 3) \& \: \mathrm{r}_{1}=2, \mathrm{r}_{2}=4}

\Rightarrow circle touch each other

\text{T.C.} \mathrm{T}=\sqrt{3} \mathrm{x}+\mathrm{y}-4=0

comparing with \mathrm{x\, \cos \theta+y \sin \theta = 2 }


\mathrm{\theta=\frac{\pi}{6} }

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Rishi

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