Given circles are

$S_1\equiv x^2+y^2-2x+18y+78=0......\left(1\right)$

Centre $C_1\equiv \left(1,-9\right)$ & radius $r_1=\sqrt{{\left(-1\right)}^2+9^2-78}=2$ units

And $x^2+y^2+8x-6y-200=0......\left(2\right)$

Centre $C_2=\left(-4,3\right)$ & radius $r_2=\sqrt{4^2+{\left(-3\right)}^2+200}=15$ units

$C_1{C}_2=\sqrt{{\left(-4-1\right)}^2+{\left(3+9\right)}^2}=13=r_2-r_1$

$\therefore$Circles touch each other internally.

Equation of common tangent to both the circles is given by $S_1-S_2=0$

$\Rightarrow x^2+y^2-2x+18y+78$$-\left(x^2+y^2+8x-6y-200\right)=0$

$\Rightarrow -10x+24y+278=0$

$\Rightarrow -5x+12y+139=0......\left(3\right)$

Option 1

$\because -5\times 0+12\times \frac{139}{12}+139=278\ne 0$

$\therefore$$\left(0,\frac{139}{2}\right)$ does not satisfy the equation $\left(3\right)$, so $\left(0,\frac{139}{2}\right)$ does not lie on the common tangent.

Option 2

 $\because -5\times \left(\frac{-137}{5}\right)+12\times \left(\frac{-1}{6}\right)+139=274\ne 0$

$\therefore \left(\frac{-137}{5},\frac{-1}{6}\right)$  does not lie on the common tangent.

Option 3

$\because -5\times 31+12\times \left(\frac{-4}{3}\right)+139=-22\ne 0$

$\therefore$$\left(31,\frac{-4}{3}\right)$ does not lie on the common tangent.

Option 4

$-5\times \left(\frac{-2}{5}\right)+12\times \left(\frac{-47}{4}\right)+139=0$

$\therefore \left(\frac{-2}{5},\frac{-47}{4}\right)$ lies on the common tangent.