The number of common tangents, to the circles x ^ 2 + y ^ 2 - 18x -...

If $S ≡ {x}^{2} + {y}^{2} + 2gx + 2fy + c = 0$.

$C ≡$\left(- g, - f\right)$and$r = \sqrt{{g}^{2} + {f}^{2} - c}$.</p><p>The given circles are${x}^{2} + {y}^{2} - 18x - 15y + 131 = 0$</p><p>$⇒ {C}_{1}\left(9, \dfrac{15}{2}\right), {r}_{1} = \sqrt{81 + \dfrac{225}{4} - 131} = \dfrac{5}{2}$</p><p>and${x}^{2} + {y}^{2} - 6x - 6y - 7 = 0$</p><p>$⇒ {C}_{2}\left(3, 3\right), {r}_{2} = \sqrt{9 + 9 + 7} = 5$</p><p>$⇒ d = {C}_{1}{C}_{2} = \sqrt{{\left(9 - 3\right)}^{2} + {\left(\dfrac{15}{2} - 3\right)}^{2}} = \sqrt{36 + \dfrac{81}{4}} = \dfrac{15}{2}$</p><p>$⇒ {r}_{1} + {r}_{2} = \dfrac{5}{2} + 5 = \dfrac{15}{2}$</p><p>$⇒ {C}_{1}{C}_{2} = {r}_{1} + {r}_{2}$</p><p>$∴$Circles touch each other externally,$3 common tangents.

Hence this is the correct option.