Question

Point $O$ is the centre of the 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 triangle $OCF$ is $2$, then the product $\left(AB\right)\left(CD\right)$ is equal to

• A. $65$
• B. $52$
• C. $78$
• D. None of the above

Answer 1

The correct option is A $65$

$a^2{e}^2=36\Rightarrow a^2-b^2=36$ ...(1)
Using $r=(s-a)\tan \frac{A}{2}$ in $△OCF$

$1=(s-a)\tan {45}^0$ when $a=CF$

$2=2(s-a)=2s-2a=2s-AB$

$=(OF+FC+CO)-AB$

$2=6+\frac{AB}{2}+\frac{CD}{2}-AB$

$\therefore \frac{AB-CD}{2}=4\Rightarrow 2(a-b)=8$

$\Rightarrow a-b=4$ ...(2)

From (1) and (2)

$a+b=9\Rightarrow 2a=13;2b=15\Rightarrow \left(AB\right)\left(CD\right)=65$