A) \[-\frac{{{a}^{2}}}{{{b}^{2}}}\]
B) \[-\frac{{{b}^{2}}}{{{a}^{2}}}\]
C) \[-\frac{{{b}^{4}}}{{{a}^{4}}}\]
D) \[-\frac{{{a}^{4}}}{{{b}^{4}}}\]
Correct Answer: D
Solution :
The equation of the chords of contact of tangents from \[({{x}_{1}},{{y}_{1}})\] and \[({{x}_{2}},{{y}_{2}})\]to the given hyperbola are\[\frac{x{{x}_{1}}}{{{a}^{2}}}-\frac{y{{y}_{1}}}{{{b}^{2}}}=1\] ...(i) and\[\frac{x{{x}_{2}}}{{{a}^{2}}}-\frac{y{{y}_{2}}}{{{b}^{2}}}=1\] ?(ii) Equation of lines (i) and (ii) are at right angled. \[\therefore \]\[\frac{{{b}^{2}}}{{{a}^{2}}}\frac{{{x}_{1}}}{{{y}_{1}}}\times \frac{{{b}^{2}}}{{{a}^{2}}}\frac{{{x}_{2}}}{{{y}_{2}}}=-1\]\[\Rightarrow \]\[\frac{{{x}_{1}}{{x}_{2}}}{{{y}_{1}}{{y}_{2}}}=-\frac{{{a}^{4}}}{{{b}^{4}}}\]
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