## BdMO National Higher Secondary 2006/12

samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### BdMO National Higher Secondary 2006/12

Two circles of radii $3$ and $6$ touches inside.Find the area of the maximum rectangle outside the circle of radius $3$ and inside the corcle of radius $6$.

samiul_samin
Posts: 1007
Joined: Sat Dec 09, 2017 1:32 pm

### Re: BdMO National Higher Secondary 2006/12

Diagram:
Screenshot_2018-03-04-15-49-27-1.png
According to the question,the radius of big circle is $6$ unit and the radius of small circle is $3$ unit.
According to the figure,$\triangle FID$ is a right triangle.Area of it will be highest if $FI=ID$
So,$FD^2=FI^2+ID^2\Rightarrow FD^2=2FI^2\Rightarrow 2FI^2=36\Rightarrow FI=3\sqrt 2$
So,$FI×ID=18$.
So,highest area of $\triangle DIF=9$
Now,we will find the area of the maximum rectangle outside the small circle and inside the big circle.Let,$[GEHF]=$Area of $GEHF$
$[GEHF]$ is highest $\Rightarrow [DIHF]$ is highest $\Rightarrow [DIF]$ is highest
Highest $[GEHF]=4[DIF]$
$\Rightarrow$ Highest$[GEHF]=36$

So, the highest area is $\fbox {36}$.