BdMO National Higher Secondary 2006/12

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samiul_samin
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BdMO National Higher Secondary 2006/12

Unread post by samiul_samin » Wed Feb 21, 2018 11:49 pm

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 $6$.

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samiul_samin
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Re: BdMO National Higher Secondary 2006/12

Unread post by samiul_samin » Sun Mar 04, 2018 4:59 pm

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 $\fbox {36}$.

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