Circle And System Of Circles Question 248
Question: The locus of centre of the circle which cuts the circles $ x^{2}+y^{2}+2g_1x+2f_1y+c_1=0 $ and $ x^{2}+y^{2}+2g_2x+2f_2y+c_2=0 $ orthogonally is
[Karnataka CET 1991]
Options:
A) An ellipse
B) The radical axis of the given circles
C) A conic
D) Another circle
Show Answer
Answer:
Correct Answer: B
Solution:
Let the circle be $ x^{2}+y^{2}+2gx+2fy+c=0 $ . This cuts the two given circles orthogonally, therefore $ 2(gg_1+ff_1)=c+c_1 $ ………….(i) and $ 2(gg_2+ff_2)=c+c_2 $ ………….(ii)
Subtracting (ii) from (i), we get
$ 2g(g_1-g_2)+2f(f_1-f_2)=c_1-c_2 $
So locus of $ (-g,\ -f) $ is $ -2x(g_1-g_2)-2y(f_1-f_2)=c_1-c_2 $ or $ 2x(g_1-g_2)+2y(f_1-f_2)+c_1-c_2=0 $ ,
which is the radical axis of the given circles.
BETA
