A) \[{{a}^{2}}co{{s}^{2}}\alpha -{{b}^{2}}si{{n}^{2}}\alpha ={{p}^{2}}\]
B) \[{{a}^{2}}co{{s}^{2}}\alpha -{{b}^{2}}si{{n}^{2}}\alpha =p\]
C) \[{{a}^{2}}co{{s}^{2}}\alpha +{{b}^{2}}si{{n}^{2}}\alpha ={{p}^{2}}c\]
D) \[{{a}^{2}}co{{s}^{2}}\alpha +{{b}^{2}}si{{n}^{2}}\alpha =p\]
E) \[{{b}^{2}}co{{s}^{2}}\alpha -{{a}^{2}}si{{n}^{2}}\alpha ={{p}^{2}}\]
Correct Answer: A
Solution :
The line\[x\text{ }cos\text{ }\alpha +y\text{ }sin\,\alpha =p\]touches the hyperbola\[\frac{{{x}^{2}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1,\]if \[{{a}^{2}}\text{ }co{{s}^{2}}\alpha -{{b}^{2}}si{{n}^{2}}\alpha ={{p}^{2}}\]
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