Circle And System Of Circles Question 315
Question: The point (2, 3) is a limiting point of a coaxial system of circles of which $ x^{2}+y^{2}=9 $ is a member. The co-ordinates of the other limiting point is given by
[MP PET 1993]
Options:
A) $ ( \frac{18}{13},\frac{27}{13} ) $
B) $ ( \frac{9}{13},\frac{6}{13} ) $
C) $ ( \frac{18}{13},-\frac{27}{13} ) $
D) $ ( -\frac{18}{13},-\frac{9}{13} ) $
Show Answer
Answer:
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=( \lambda ,\ \frac{3\lambda }{2} )\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 $ [ \frac{18}{13},\ \frac{3}{2}( \frac{18}{13} ) ] $ or $ ( \frac{18}{13},\ \frac{27}{13} ) $ .
BETA
