The locus of the center of the circle which c | vedclass

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(B) Let the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$. This circle cuts the two given circles orthogonally.The condition for two circles $x^2 + y^2 +

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The radical axis of the given circles

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(B) Let the circle be $x^2 + y^2 + 2gx + 2fy + c = 0$. This circle cuts the two given circles orthogonally.The condition for two circles $x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0$ and $x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0$ to cut orthogonally is $2g_1g_2 + 2f_1f_2 = c_1 + c_2$.Applying this to our circle,we get:$2gg_1 + 2ff_1 = c + c_1$ .... $(i)$$2gg_2 + 2ff_2 = c + c_2$ .... $(ii)$Subtracting $(ii)$ from $(i)$,we get:$2g(g_1 - g_2) + 2f(f_1 - f_2) = c_1 - c_2$The center of the circle is $(-g, -f)$. Let the center be $(x, y)$,so $g = -x$ and $f = -y$.Substituting these into the equation:$2(-x)(g_1 - g_2) + 2(-y)(f_1 - f_2) = c_1 - c_2$$-2x(g_1 - g_2) - 2y(f_1 - f_2) = c_1 - c_2$$2x(g_1 - g_2) + 2y(f_1 - f_2) + c_1 - c_2 = 0$This is the equation of the radical axis of the two given circles.

The locus of the center 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

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