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 $[\sqrt{k}] =$ ______. Where [.] denotes greatest integer function

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answer is 8.

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Detailed Solution

$\begin{array}{l} r = inradius of ΔOCF = 1 \\ OF = 6 and r = 1 \Rightarrow Δ = s \\ \frac{1}{2} OF . OC = s \Rightarrow 6 b = 2 s \\ (OC = b) (OF = ae = 6) \end{array}$

Perimeter of $ΔOCF = 2 s \Rightarrow 6 + b + a = 6 b$
$\begin{array}{l} \Rightarrow 5 b = a + 6 \\ \Rightarrow a^{2} - a^{2} e^{2} = b^{2} \Rightarrow b = \frac{5}{2} \\ ∴ a = \frac{13}{2}; K = AB \cdot CD = 2 a \cdot 2 b \\ [\sqrt{K}] = [\sqrt{65}] = 8 \end{array}$