Four tangents

Geometry Level 4

Given two circles with equations \((x+5)^2+(y+2)^2=16\) and \((x-3)^2+(y-4)^2=25\), there are four tangents to them, at the same time.

If two of them are: \[y=\dfrac{a \pm b\sqrt{c}}{d}x+\dfrac{e \pm f\sqrt{c}}{d}\] And the other two are: \[y=\dfrac{g \pm h\sqrt{j}}{k}x+\dfrac{l \pm m\sqrt{j}}{k}\]

Find \(a+b+c+d+e+f+g+h+j+k+l+m\), where the equations of the line are in its simplest form.

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