A) Length of AB is constant
B) PA and PB are always equal
C) The locus of the midpoint of AB is \[{{x}^{2}}+{{y}^{2}}={{x}^{2}}{{y}^{2}}\]
D) None of these
Correct Answer: C
Solution :
Let \[P({{x}_{1}},\ {{y}_{1}})\]be a point on\[{{x}^{2}}+{{y}^{2}}=4\]. Then the equation of the tangent at P is \[x{{x}_{1}}+y{{y}_{1}}=4\] which meets the coordinate axes at \[A\left( \frac{4}{{{x}_{1}}},\ 0 \right)\] and\[B\text{ }\left( 0,\ \frac{4}{{{y}_{1}}} \right)\]. Obviously, (a) and (b) are not true. Let \[(h,\ k)\] be the mid-point of AB. Therefore \[h=\frac{2}{{{x}_{1}}},\ k=\frac{2}{{{y}_{1}}}\] i.e., \[{{x}_{1}}=\frac{2}{h},\ {{y}_{1}}=\frac{2}{k}\] But \[({{x}_{1}},\ {{y}_{1}})\] lies on\[{{x}^{2}}+{{y}^{2}}=4\].
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