Let $r_1$ and $r_2$ represent the radii of the smallest and the largest circle passing through $(a, \frac {1}{a})$ and touching the circle $$4x^2+4y^2-4x-8y-31=0$$ where $a$ is the least possible value of the ratio of sides of the regular $n$ sided polygon($n\ge 3$) inscribed and circumscribed to a unit circle then find the the value of $r_1+r_2$
My work:-
The side of the n sided polygon whose inradius is $1$ is $$\tan {\frac {\pi}{n}}$$ and the side of $n$ sided polygon if the circumradius is $1$ is $$\sin {\frac {\pi}{n}}$$ For the value of $a$ we need to find the minimum value of $$\sec \frac{\pi}{n}$$ Now for the circle to be the largest one, the given circle must touch it internally while the smallest circle must touch it externally. But using this logic I am getting the wrong answer. Can someone please provide a bit better approach to such problem.
$$ {\text{side of the $n$-sided regular polygon inscribed in a unit circle} \over \text{side of the $n$-sided regular polygon circumscribed to a unit circle}} =\cos{\pi\over n}, $$ the minimum $a={1\over2}$ being attained for $n=3$.
As the given circle has center $\big({1\over2},1\big)$ and radius $3$, the smallest and the largest circle passing through $(a,1/a)$ and touching the circle have diameters of $2$ and $4$ respectively.