# Where do I look at this from? (2)

Geometry Level 5

Consider a circle $$S_{1}$$,

$x^{2}+y^{2}+1-2x(\cos\theta - \sin\theta)-2y(\cos\theta + \sin\theta)=0$

Now, consider another circle $$S_{2}$$, centred at $$P=(-\cos\theta - \sin\theta, \cos\theta - \sin\theta)$$ such that $$S_{1}$$ internally touches $$S_{2}$$.

Draw a pair of tangents $$T_{1}$$ and $$T_{2}$$ from $$P$$ to $$S_{1}$$. Let these tangents meet the circle $$S_{2}$$ at points $$A$$ and $$B$$ as shown. From a point $$R$$ on $$S_{2}$$, draw chords $$RA$$ and $$RB$$ with lengths $$l_{1} , l_{2}$$, respectively to $$S_{2}$$.

Then, there exists a certain $$\alpha$$ such that one of

$\large \left| \cos^{-1}\frac{l_{1}}{6} + \cos^{-1}\frac{l_{2}}{6}\right| \quad \text{ or } \quad \left|\cos^{-1}\frac{l_{1}}{6}-\cos^{-1}\frac{l_{2}}{6} \right|$

is equal to $$\alpha$$ regardless of the value of $$R$$. Find the value of this constant $$\alpha$$.

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