A) \[\frac{2b}{\sqrt{{{a}^{2}}-4{{b}^{2}}}}\]
B) \[\frac{\sqrt{{{a}^{2}}-4{{b}^{2}}}}{2b}\]
C) \[\frac{2b}{a-2b}\]
D) \[\frac{b}{a-2b}\]
Correct Answer: A
Solution :
Any tangent to \[{{x}^{2}}+{{y}^{2}}={{b}^{2}}\] is \[y=mx-b\,\sqrt{1+{{m}^{2}}}.\]It touches\[{{(x-a)}^{2}}+{{y}^{2}}={{b}^{2}}\], if \[\frac{ma-b\sqrt{1+{{m}^{2}}}}{\sqrt{{{m}^{2}}+1}}=b\] or \[ma=2b\sqrt{1+{{m}^{2}}}\] or\[{{m}^{2}}{{a}^{2}}=4{{b}^{2}}+4{{b}^{2}}{{m}^{2}}\], \[\therefore \]\[m=\pm \,\frac{2b}{\sqrt{{{a}^{2}}-4{{b}^{2}}}}\].
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