If $$x^2 + y^2 - 2x + 6y + 10 = 0$$, then the value of $$(x^2 + y^2)$$ is

$$x^2\ +\ y^2-2x\ +\ 6y+10=0$$

$$x^2\ -2x+\ y^2\ +\ 6y+9+1=0$$

$$x^2\ -2x+1+\ y^2\ +\ 6y+9=0$$

$$\left(x-1\right)^2+\ \left(y+3\right)^2=0$$

Sum of two squares is zero hence both the squares should be zero

Therefore, $$\left(x-1\right)^2=0$$ and $$\left(y+3\right)^2=0$$

                                   x = 1 and  y = -3

        $$=$$>     $$x^2+y^2=\left(1\right)^2+\left(-3\right)^2$$

                                  $$=10$$