A) 3
B) 4
C) 5
D) 6
Correct Answer: B
Solution :
[b] Perpendicular tangents intersect on the director circle. The director circle of \[\frac{{{x}^{2}}}{2}+\frac{{{y}^{2}}}{1}=1\]is\[{{x}^{2}}+{{y}^{2}}=3\]. Let any point on this circle be \[P(\sqrt{3}\cos \theta ,\sqrt{3}\sin \theta )\] Chord of contact is \[x\sqrt{3}\,\cos \theta +2y\sqrt{3}\sin \theta -2=0.\]. If it touches \[{{x}^{2}}+{{y}^{2}}={{r}^{2}},\]then \[r=\frac{2}{\sqrt{3{{\cos }^{2}}\theta +12{{\sin }^{2}}\theta }}=\frac{2}{\sqrt{3+9{{\sin }^{2}}\theta }}\] \[{{r}_{\max }}=\frac{2}{\sqrt{3}}\] and \[{{r}_{\min }}=\frac{2}{\sqrt{12}}\] So, \[\frac{{{A}_{\max }}}{{{A}_{\min }}}=4\]
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