If the form any point on the circle $x^{2} + y^{2} = a^{2}$ , tangents are drawn to the circle $x^{2} + y^{2} = b^{2} (a > b)$ , then the angle between the tangents is $k {sin}^{- 1} \frac{b}{a}$ , for some integer $k$ . Find that integer.

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Solution

Let $P$ be any point on $x^{2} + y^{2} = a^{2}$ .
Let $P T, P T^{'}$ be the tangent to the concentric circle.
$x^{2} + y^{2} = b^{2}$
Join $O P, O T$ and $O T^{'}$ .
Then $∠ O P T = ∠ O P T^{'} = θ$ (say)
and $O T ⊥ P T$
Also, $O T = b$ and $O P = a$ .
From right-angles triangle $O P T$ , $sin θ = \frac{b}{a} \Rightarrow θ = {sin}^{- 1} \frac{b}{a}$
$∴$ Angle between the tangents $= 2 θ = 2 {sin}^{- 1} \frac{b}{a} \Rightarrow k = 2$

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