Circle cuts circle - (4)

Geometry Level 5

Two circles of radii one and two units respectively cut each other at an angle of 6 0 ∘ 60^\circ . If the length of the common chord of these two circles can be expressed in the form a b c \dfrac{a\sqrt{b}}{c} where b b is square free , find the value of a + b + c a+b+c .

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The answer is 30.

Nihar Mahajan by Nihar Mahajan , 20, India

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2 solutions


A p p l y i n g S i n L a w t o Δ A B C , A B S i n ( 60 − α ) = B C S i n ( α ) ⟹ 1 2 = S i n 60 ∗ C o s α − C o s 60 ∗ S i n α S i n α . S i m p l i f y i n g C o t α = 2 3 . ∴ S i n α = 21 7 . C o m m o n C h o r d = 2 ∗ B D = 2 ∗ A B S i n α = 2 21 7 . a + b + c = 2 + 21 + 7 = 30 Applying ~Sin~ Law ~to ~\Delta~ ABC , ~\dfrac{AB}{Sin(60-\alpha)} =\dfrac{BC}{Sin(\alpha)} \\ \implies ~\dfrac 1 2=\dfrac{Sin60*Cos\alpha - Cos60*Sin\alpha}{Sin\alpha}. \\ Simplifying ~ Cot\alpha=\dfrac 2 {\sqrt3}. ~~\therefore ~Sin\alpha=\dfrac {\sqrt {21}} 7.\\ Common~ Chord=2*BD=2*ABSin\alpha=2\dfrac {\sqrt {21}} 7.~~~~~~a+b+c=2+21+7=30

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