Hyperbola 3 Question 1
1. If $x=9$ is the chord of contact of the hyperbola $x^{2}-y^{2}=9$, then the equation of the corresponding pair of tangents is
$(1999,2 M)$
(a) $9 x^{2}-8 y^{2}+18 x-9=0$
(b) $9 x^{2}-8 y^{2}-18 x+9=0$
(c) $9 x^{2}-8 y^{2}-18 x-9=0$
(d) $9 x^{2}-8 y^{2}+18 x+9=0$
Objective Question II
(Only one or more than one correct option)
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Answer:
Correct Answer: 1. (b)
Solution:
- Let $(h, k)$ be a point whose chord of contact with respect to hyperbola $x^{2}-y^{2}=9$ is $x=9$.
We know that, chord of contact of $(h, k)$ with respect to hyperbola $x^{2}-y^{2}=9$ is $T=0$.
$\Rightarrow h \cdot x+k(-y)-9=0$
$\therefore \quad h x-k y-9=0$
But it is the equation of the line $x=9$.
This is possible when $h=1, k=0$ (by comparing both equations).
Again equation of pair of tangents is
$$ \begin{array}{lc} & T^{2}=S S _1 \\ \Rightarrow & (x-9)^{2}=\left(x^{2}-y^{2}-9\right)\left(1^{2}-0^{2}-9\right) \\ \Rightarrow & x^{2}-18 x+81=\left(x^{2}-y^{2}-9\right)(-8) \\ \Rightarrow & x^{2}-18 x+81=-8 x^{2}+8 y^{2}+72 \\ \Rightarrow & 9 x^{2}-8 y^{2}-18 x+9=0 \end{array} $$
