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Q. Which of the following statement is(are) true?

Conic Sections

Solution:

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(A) Let the point be $\left(\frac{ a }{ e }, \beta\right)$.
Chord of contact is $T =0$
i.e., $x\left(\frac{a}{e}\right)+y \beta=a^2$, which is passing through ( $\left.a e, 0\right)$.
(B) $ T \left( at _1 t _2, a \left( t _1+ t _2\right)\right) \Rightarrow at _1 t _2= a \Rightarrow t _1 t _2=1$
[Hence equation of PQ is $2 x-\left(t_1+t_2\right) y+2 a=0$ $(x+a)-\lambda y=0]$
$\Rightarrow $ chord PQ passes $(- a , 0)$ which is foot of directrix $\therefore $ (B) is also true.
(C) $ As \left| PS _1- PS _2\right|=$ constant so, for hyperbola
$0< k ^2+1<5 $
$\Rightarrow k ^2<4 $
$\Rightarrow-2< k <2$
So, number of integral values of $k$ are 3 (i.e., $k =-1,0,1$ )