The radical axis of the circles S 1 : x 2 + y 2 - 4 x + 6 y - 10 = 0 and S 2 : x 2 + y 2 + 2 x - 6 y + 2 = 0…

Given: $S_1: x^2+y^2-4x+6y-10=0$

$S_2: x^2+y^2+2x-6y+2=0$

$r=\sqrt{g^2+f^2-c} c\equiv \left(-g,-f\right)$

$r_1=\sqrt{2^2+{\left(-3\right)}^2+10}=\sqrt{23}$

$c_1\equiv \left(-g,-f\right)\equiv \left(2,-3\right)$

$r_2=\sqrt{{\left(-1\right)}^2+{\left(3\right)}^2-2}=\sqrt{1+9-2}=\sqrt{8}=2\sqrt{2}$

$c_2\equiv \left(-1,3\right)$

$c_1{c}_2=\sqrt{{\left(2+1\right)}^2+{\left(-3-3\right)}^2}$

$c_1{c}_2=\sqrt{9+36}$

$c_1{c}_2=\sqrt{45}$

$c_1{c}_2=3\sqrt{5}$

$r_1+r_2=\sqrt{23}+2\sqrt{2}$

Here, $c_1{c}_2<r_1+r_2$

So radical axis will cut circles $S_1 \& S_2$ at two real and distinct points.