Point “O” is the centre of an ellipse with major axis AB and minor axis CD. Point F is one focus of the ellipse. If OF = 6 and the diameter of the inscribed circle of the triangle OCF is 2 and the product of (AB)(CD) is k. Then the value of[k]=______. Where [.] denotes greatest integer function

# Point “O” is the centre of an ellipse with major axis AB and minor axis CD. Point F is one focus of the ellipse. If OF = 6 and the diameter of the inscribed circle of the triangle OCF is 2 and the product of (AB)(CD) is k. Then the value of$\left[\sqrt{\mathrm{k}}\right]=$______. Where [.] denotes greatest integer function

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### Solution:

Perimeter of$\mathrm{\Delta OCF}=2\mathrm{s}⇒6+\mathrm{b}+\mathrm{a}=6\mathrm{b}$
$\begin{array}{l}⇒5\mathrm{b}=\mathrm{a}+6\\ ⇒{\mathrm{a}}^{2}-{\mathrm{a}}^{2}{\mathrm{e}}^{2}={\mathrm{b}}^{2}⇒\mathrm{b}=\frac{5}{2}\\ \therefore \mathrm{a}=\frac{13}{2};\mathrm{K}=\mathrm{AB}\cdot \mathrm{CD}=2\mathrm{a}\cdot 2\mathrm{b}\\ \left[\sqrt{\mathrm{K}}\right]=\left[\sqrt{65}\right]=8\end{array}$  +91

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