A) \[\frac{2xy}{{{x}^{2}}-{{y}^{2}}}\]
B) \[\frac{2xy}{{{x}^{2}}+{{y}^{2}}}\]
C) \[\frac{{{x}^{2}}+{{y}^{2}}}{2xy}\]
D) \[\frac{{{x}^{2}}-{{y}^{2}}}{2xy}\]
Correct Answer: A
Solution :
[a] Given, \[\cos \theta =\frac{{{x}^{2}}-{{y}^{2}}}{{{x}^{2}}+{{y}^{2}}}\] In right angled \[\Delta ABC,\]\[{{(BC)}^{2}}={{(AC)}^{2}}-{{(AB)}^{2}}\] \[{{(BC)}^{2}}=[{{({{x}^{2}}+{{y}^{2}})}^{2}}-{{({{x}^{2}}-{{y}^{2}})}^{2}}]\] \[{{(BC)}^{2}}=({{x}^{2}}+{{y}^{2}}+{{x}^{2}}-{{y}^{2}})({{x}^{2}}+{{y}^{2}}-{{x}^{2}}+{{y}^{2}})\] \[{{(BC)}^{2}}=2{{x}^{2}}\times 2{{y}^{2}}=4{{x}^{1}}{{y}^{2}},\] \[\therefore \]\[\cot \theta =\frac{AB}{BC}=\frac{{{x}^{2}}-{{y}^{2}}}{2xy}\] |
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