A) \[0\]
B) 1
C) \[4\]
D) infinite
Correct Answer: A
Solution :
[a] Normal to the given hyperbola at any point \[(a\sec \theta ,b\tan \theta )\] is \[ax\cos \theta +by\cot \theta ={{a}^{2}}+{{b}^{2}}\] ...(1) Normal to the conjugate hyperbola at any point \[(a\tan \phi ,b\,sec\phi )\]is \[ax\,\cot \phi +by\,\cos \phi ={{a}^{2}}+{{b}^{2}}\] ?.(2) If (1) and (2) represent the same line, then \[\frac{\cot \phi }{\cos \theta }=\frac{\cos \phi }{\cot \theta }=1\] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\tan \phi =\sec \theta \] and \[\sec \phi =\tan \theta \] \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,{{\sec }^{2}}\theta -{{\tan }^{2}}\theta ={{\sec }^{2}}\phi -{{\tan }^{2}}\phi ,\]which is not possible. Hence, there is no common normal.
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