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Two circles with radii $a$ and $b$ touch each other externally such that $θ$ is the angle between the direct common tangents $(a > b \geq 2) .$ Then

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$θ = 2 \cos^{- 1} (\frac{a - b}{a + b})$

$θ = 2 \tan^{- 1} (\frac{a + b}{a - b})$

$θ = 2 \sin^{- 1} (\frac{a + b}{a - b})$

$θ = 2 \sin^{- 1} (\frac{a - b}{a + b})$

answer is D.

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Detailed Solution

$\begin{array}{l} \sin α = \frac{a - b}{a + b} \\ α = \sin^{- 1} (\frac{a - b}{a + b}) \end{array}$

angle $2 α = 2 \sin^{- 1} (\frac{a - b}{a + b})$

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