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

\frac{5 π}{6}

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\frac{2 π}{3}

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\frac{π}{3}

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\frac{π}{6}

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Solution

The correct option is D $\frac{π}{6}$
$C_{1} C_{2} = r_{1} + r_{2}$
$C_{1} = (0, 0); C_{2} = (3 \sqrt{3}, 3)$
and $r_{1} = 2, r_{2} = 4$
$\Rightarrow$ Circle touch each other externally.
Equation of common tangent is $\sqrt{3} x + y - 4 = 0$ ...... (i)
Comparing it with $x cos θ + y sin θ = 2$ , we get $θ = \frac{π}{6}$

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