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10-1.Circle and System of Circles
normal
The locus of centre of the circle which cuts the circles${x^2} + {y^2} + 2{g_1}x + 2{f_1}y + {c_1} = 0$ and ${x^2} + {y^2} + 2{g_2}x + 2{f_2}y + {c_2} = 0$ orthogonally is
A
An ellipse
B
The radical axis of the given circles
C
A conic
D
Another circle
Solution
(b) Let the circle be ${x^2} + {y^2} + 2gx + 2fy + c = 0$. This cuts the two given circles orthogonally,
therefore $2(g{g_1} + f{f_1}) = c + {c_1}$….(i)
and $2(g{g_2} + f{f_2}) = c + {c_2}$….(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.
Standard 11
Mathematics
