A) \[\frac{9}{4}\]
B) \[-\frac{9}{4}\]
C) \[\frac{81}{16}\]
D) \[-\frac{81}{16}\]
Correct Answer: D
Solution :
The equation of hyperbola is \[4{{x}^{2}}-9{{y}^{2}}=36\] \[\Rightarrow \] \[\frac{{{x}^{2}}}{9}-\frac{{{y}^{2}}}{4}=1\] ?.(i) The equation of the chord of contact of cangents from \[({{x}_{1}},{{y}_{1}})\]and\[({{x}_{2}},{{y}_{2}})\] to the given hyperbola are \[\frac{x{{x}_{1}}}{9}-\frac{y{{y}_{1}}}{4}=1\] ?.(ii) and \[\frac{x{{x}_{2}}}{9}-\frac{y{{y}_{2}}}{4}=1\] ?(iii) Lines (ii) and (m) are at right angles \[\therefore \] \[\frac{4}{9}.\frac{{{x}_{1}}}{{{y}_{1}}}\times \frac{4}{9}.\frac{{{x}_{2}}}{{{y}_{2}}}=-1\]\[(\because \,{{m}_{1}}{{m}_{2}}=-1)\] \[\Rightarrow \] \[\frac{{{x}_{1}}{{x}_{2}}}{{{y}_{1}}{{y}_{2}}}=-{{\left( \frac{9}{4} \right)}^{2}}\] \[=-\frac{81}{16}\]
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