From which point are the lengths of the tange | vedclass

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(C) The point from which the lengths of tangents to three circles are equal is the radical center of the three circles.Let the equations be:$S_1: x^2

Vedclass · Accepted Answer

$\left( 8, -\frac{15}{2} \right)$

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(C) The point from which the lengths of tangents to three circles are equal is the radical center of the three circles.Let the equations be:$S_1: x^2 + y^2 - 8x + 40 = 0$$S_2: x^2 + y^2 - 5x + 16 = 0$ (Dividing by $5$)$S_3: x^2 + y^2 - 8x + 16y + 160 = 0$Radical axis of $S_1$ and $S_2$ is $S_1 - S_2 = 0 \implies (-8x + 5x) + (40 - 16) = 0 \implies -3x + 24 = 0 \implies x = 8$.Radical axis of $S_1$ and $S_3$ is $S_1 - S_3 = 0 \implies (-8x + 8x) - 16y + (40 - 160) = 0 \implies -16y - 120 = 0 \implies y = -\frac{120}{16} = -\frac{15}{2}$.Thus,the radical center is $\left( 8, -\frac{15}{2} \right)$.

From which point are the lengths of the tangents drawn to the circles $x^{2} + y^{2} - 8x + 40 = 0$,$5x^{2} + 5y^{2} - 25x + 80 = 0$,and $x^{2} + y^{2} - 8x + 16y + 160 = 0$ equal?

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