A) \[\left( \frac{18}{13},\frac{27}{13} \right)\]
B) \[\left( \frac{9}{13},\frac{6}{13} \right)\]
C) \[\left( \frac{18}{13},-\frac{27}{13} \right)\]
D) \[\left( -\frac{18}{13},-\frac{9}{13} \right)\]
Correct Answer: A
Solution :
\[{{(x-2)}^{2}}+{{(y-3)}^{2}}=0\] or \[({{x}^{2}}+{{y}^{2}}-9)-4x-6y+22=0\] or \[({{x}^{2}}+{{y}^{2}}-9)-\lambda (2x+3y-11)=0\] represents the family of co-axial circles. \[C=\left( \lambda ,\ \frac{3\lambda }{2} \right)\text{ },\ r=\sqrt{{{\lambda }^{2}}+\frac{9{{\lambda }^{2}}}{4}-11\lambda +9}\] For limiting points \[r=0\] \[\Rightarrow 13{{\lambda }^{2}}-44\lambda +36=0\Rightarrow \lambda =\frac{18}{13},\ 2\] \[\therefore \]The limiting points are (2, 3) and \[\left[ \frac{18}{13},\ \frac{3}{2}\left( \frac{18}{13} \right) \right]\] or \[\left( \frac{18}{13},\ \frac{27}{13} \right)\].
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