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

see full answer

Talk to JEE/NEET 2025 Toppers - Learn What Actually Works!

Real Strategies. Real People. Real Success Stories - Just 1 call away

An Intiative by Sri Chaitanya

answer is 8.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

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