Question

Let $r_1$ and $r_2$ represent the radii of the smallest and largest circles 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 ratio of the sides of the regular $n$ sided $(n\ge 3)$ polygons inscribed and circumscribed to a unit circle, then $r_1+r_2$ is

• A. $6$
• B. $5$
• C. $4$
• D. $3$

Answer 1

The correct option is D $3$

We have ratio of $n$ sided regular polygon inside and outside unit circle $=\frac{2\sin \frac{\pi }{n}}{2\tan \frac{\pi }{n}}=\cos \frac{\pi }{n};$

Which is least when $n=3$ $(asn\ge 3)\Rightarrow a=\frac{1}{2}$
Now,

$\left(\frac{1}{2},2\right)$ lies inside the circle with centre $\left(\frac{1}{2},1\right)$ and radius $3$;

Sum of radii of smallest and largest circle is $R$
also $2(r_1+r_2)=2R\Rightarrow r_1+r_2=3$