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(A) The equation of the line passing through the origin is $y = mx$,which can be written as $mx - y = 0$.The circle is given by $(x - 4)^2 + (y + 5)^2

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(A) The equation of the line passing through the origin is $y = mx$,which can be written as $mx - y = 0$.The circle is given by $(x - 4)^2 + (y + 5)^2 = 25$,which has center $(4, -5)$ and radius $r = 5$.For the line to be tangent to the circle,the perpendicular distance from the center $(4, -5)$ to the line $mx - y = 0$ must be equal to the radius $r = 5$.Using the distance formula: $\frac{|m(4) - (-5)|}{\sqrt{m^2 + (-1)^2}} = 5$.$|4m + 5| = 5\sqrt{m^2 + 1}$.Squaring both sides: $(4m + 5)^2 = 25(m^2 + 1)$.$16m^2 + 40m + 25 = 25m^2 + 25$.$9m^2 - 40m = 0$.$m(9m - 40) = 0$.Thus,$m = 0$ or $m = \frac{40}{9}$.

If a line passing through the origin touches the circle $(x - 4)^2 + (y + 5)^2 = 25$,then its slope is:

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