to #2:
1. Use two different values of k: k = r and k = s
2. You'll get two different circles:
$\displaystyle A: x^2 + y^2 + r\cdot x + (1 + r)\cdot y - (1 + r) = 0$
$\displaystyle B: x^2 + y^2 + s\cdot x + (1 + s)\cdot y - (1 + s) = 0$
2. Calculate the coordinates of the common points (points of intersection) by subtracting columnwise:
$\displaystyle A-B: x(r - s) + y(r - s) - (r - s) = 0$
Divide by (r-s):
$\displaystyle x+y-1=0~\implies~x=1-y$
3. Plug in this term for x into the original equation:
$\displaystyle (1 - y)^2 + y^2 + k(1 - y) + (1 + k)y - (1 + k) = 0~\implies~ 2y^2 - y = 0$
Solve for y: $\displaystyle y=0~\vee~y=\dfrac12$
4. Calculate the x-coordinate. You'll get 2 points whose coordinates are not depending on k, that means each circle passes through these two points: $\displaystyle P_1(1,0)\ ,\ P_2\left(\dfrac12\ ,\ \dfrac12\right)$