Find the radical center of the three circles | vedclass

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(C) Let the equations of the circles be:$S_1: x^2 + y^2 - a^2 = 0$$S_2: x^2 + y^2 - 2cx + c^2 - a^2 = 0$$S_3: x^2 + y^2 - 2by + b^2 - a^2 = 0$The radi

Vedclass · Accepted Answer

$(c/2, b/2)$

Vedclass · Answer

(C) Let the equations of the circles be:$S_1: x^2 + y^2 - a^2 = 0$$S_2: x^2 + y^2 - 2cx + c^2 - a^2 = 0$$S_3: x^2 + y^2 - 2by + b^2 - a^2 = 0$The radical axis of $S_1$ and $S_2$ is given by $S_1 - S_2 = 0$:$(x^2 + y^2 - a^2) - (x^2 + y^2 - 2cx + c^2 - a^2) = 0$$2cx - c^2 = 0 \implies x = c/2$The radical axis of $S_1$ and $S_3$ is given by $S_1 - S_3 = 0$:$(x^2 + y^2 - a^2) - (x^2 + y^2 - 2by + b^2 - a^2) = 0$$2by - b^2 = 0 \implies y = b/2$The radical center is the intersection of these radical axes,which is $(c/2, b/2)$.

Find the radical center of the three circles $x^2 + y^2 = a^2$,$(x - c)^2 + y^2 = a^2$,and $x^2 + (y - b)^2 = a^2$.

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