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ln a$\Delta A B C,$ if $\angle C=90^{\circ}, r$ and $R$ are the inradius and circumrodius of the ABC respectively, then $2(r+R)$ is equal to
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The correct answer is:
$a+b$
Tangents drawn from external points are of equal length.
$\Rightarrow \quad B D=B E=a-r$
$[\because C D=r]$
and $\quad A F=A E=b-r$
$[\because C F=r]$
$A E+B E=A B$
$\Rightarrow \quad b-r+a-r=2 R$
$1 \because A B$ is a diameter of circumcircle]
$\Leftrightarrow \quad b+a=2 R+2 r$
$\Rightarrow \quad 2(r+R)=a+b$
$\Rightarrow \quad B D=B E=a-r$
$[\because C D=r]$
and $\quad A F=A E=b-r$
$[\because C F=r]$
$A E+B E=A B$
$\Rightarrow \quad b-r+a-r=2 R$
$1 \because A B$ is a diameter of circumcircle]
$\Leftrightarrow \quad b+a=2 R+2 r$
$\Rightarrow \quad 2(r+R)=a+b$
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