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(D) The radical centre is the point $P$ from which the lengths of the tangents drawn to the circles are equal. Let the circles be $S_1: x^2+y^2-4=0$,$

Vedclass · Accepted Answer

$(5, 2)$

Vedclass · Answer

(D) The radical centre is the point $P$ from which the lengths of the tangents drawn to the circles are equal. Let the circles be $S_1: x^2+y^2-4=0$,$S_2: x^2+y^2-2x+3y=0$,and $S_3: x^2+y^2+7y-18=0$.The radical axis of $S_1$ and $S_2$ is given by $S_1-S_2=0$:$(x^2+y^2-4)-(x^2+y^2-2x+3y)=0$$2x-3y-4=0$ ... $(i)$The radical axis of $S_1$ and $S_3$ is given by $S_1-S_3=0$:$(x^2+y^2-4)-(x^2+y^2+7y-18)=0$$-7y+14=0$$7y=14 \Rightarrow y=2$Substituting $y=2$ in equation $(i)$:$2x-3(2)-4=0$$2x-6-4=0$$2x=10 \Rightarrow x=5$Thus,the radical centre $P$ is $(5, 2)$.

If the lengths of the tangents drawn from a point $P$ to the three circles $x^2+y^2-4=0$,$x^2+y^2-2x+3y=0$,and $x^2+y^2+7y-18=0$ are equal,then the coordinates of $P$ are

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