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Point $' O'$ is the Centre of the ellipse with major axis $A B$ & minor axis $C D$ . Point $F$ is one focus of the ellipse. If $O F = 6$ & the diameter of the inscribed circle of triangle $O C F$ is 2, then the product $(A B) (C D)$ is equal to

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Detailed Solution

$a^{2} e^{2} = 36 \Rightarrow a^{2} - b^{2} = 36.... (1); 4 a b = ?$

Using $r = (s - a) \tan \frac{A}{2}$ in $Δ O C F$

$1 = (s - a) \tan 45$ where $a = C F$

$2 = 2 (s - a)$

Or $2 = 2 s - 2 a = 2 s - A B$

Or $2 = (O F + F C + C O) - A B$

$2 = 6 + \frac{A B}{2} + \frac{C D}{2} - A B$

$\frac{A B - C D}{2} = 4 \Rightarrow 2 (a - b) = 8 \Rightarrow a - b = 4..... (2)$

From $(1) & (2) a + b = 9 \Rightarrow 2 a = 13;$

$2 b = 5 \Rightarrow (A B) (C D) = 65$

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