A) \[(-1,2),\left( \frac{3}{5},\frac{-14}{5} \right)\]
B) \[(-1,2),\left( \frac{3}{5},\frac{14}{5} \right)\]
C) \[(-1,2),\left( \frac{-3}{5},\frac{14}{5} \right)\]
D) None of these
Correct Answer: B
Solution :
| [b] The equation of two circles are |
| \[{{x}^{2}}+{{y}^{2}}-2x-6y+9=0\] ?. (1) |
| and \[{{x}^{2}}+{{y}^{2}}+6x-2y+1=0\] ?. (2) |
| Their radical axis is |
| \[8x+4y-8=0\] or \[2x+y-2=0\] ? (3) |
| The equation of any circle coaxial with the given circles is |
| \[{{x}^{2}}+{{y}^{2}}-2x-6y+9+\lambda (2x+y-2)=0\] |
| or \[{{x}^{2}}+{{y}^{2}}+(2\lambda -2)x+(\lambda -6)y+(9-2\lambda )=0\] ?. (4) |
| The centre of this circle is \[[(1-\lambda ),\,\,\left( 3-\frac{\lambda }{2} \right)]\] |
| Its radius\[=\sqrt{{{(1-\lambda )}^{2}}+{{\left( 3-\frac{\lambda }{2} \right)}^{2}}-(9-2\lambda )}\] |
| \[=\sqrt{\frac{5{{\lambda }^{2}}}{4}-3\lambda +1}\] |
| For limiting points its radius = 0 |
| i.e., \[\frac{5{{\lambda }^{2}}}{4}-3\lambda +1=0\] or \[5{{\lambda }^{2}}-12\lambda +4=0\therefore \] |
| \[\lambda =2,\,\,\frac{2}{5}\] |
| Substituting these values in (5), the limiting points are |
| \[(-1,2)\] and \[\left( \frac{3}{5},\frac{14}{5} \right)\] |
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