Point $'O' $ is the centre of the ellipse with major axis $AB$ $ \&$ minor axis $CD$. Point $F$ is one focus of the ellipse. If $OF = 6 $ $ \&$ the diameter of the inscribed circle of triangle $OCF$ is $2, $ then the product $ (AB)\,(CD) $ is equal to
Point $'O' $ is the centre of the ellipse with major axis $AB$ $ \&$ minor axis $CD$. Point $F$ is one focus of the ellipse. If $OF = 6 $ $ \&$ the diameter of the inscribed circle of triangle $OCF$ is $2, $ then the product $ (AB)\,(CD) $ is equal to
- A
$65$
- B
$52$
- C
$78$
- D
none
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Define the collections $\left\{ E _1, E _2, E _3, \ldots ..\right\}$ of ellipses and $\left\{ R _1, K _2, K _3, \ldots ..\right\}$ of rectangles as follows : $E_1: \frac{x^2}{9}+\frac{y^2}{4}=1$
$K _1$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _1$;
$E_n$ : ellipse $\frac{x^2}{a_n^2}+\frac{y^2}{b_{n}^2}=1$ of largest area inscribed in $R_{n-1}, n>1$;
$R _{ n }$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _{ n }, n >1$.
Then which of the following options is/are correct?
$(1)$ The eccentricities of $E _{18}$ and $E _{19}$ are NOT equal
$(2)$ The distance of a focus from the centre in $E_9$ is $\frac{\sqrt{5}}{32}$
$(3)$ The length of latus rectum of $E_Q$ is $\frac{1}{6}$
$(4)$ $\sum_{n=1}^N\left(\right.$ area of $\left.R_2\right)<24$, for each positive integer $N$
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$K _1$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _1$;
$E_n$ : ellipse $\frac{x^2}{a_n^2}+\frac{y^2}{b_{n}^2}=1$ of largest area inscribed in $R_{n-1}, n>1$;
$R _{ n }$ : rectangle of largest area, with sides parallel to the axes, inscribed in $E _{ n }, n >1$.
Then which of the following options is/are correct?
$(1)$ The eccentricities of $E _{18}$ and $E _{19}$ are NOT equal
$(2)$ The distance of a focus from the centre in $E_9$ is $\frac{\sqrt{5}}{32}$
$(3)$ The length of latus rectum of $E_Q$ is $\frac{1}{6}$
$(4)$ $\sum_{n=1}^N\left(\right.$ area of $\left.R_2\right)<24$, for each positive integer $N$
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