Circle COMPLICATED problem

[Kaworito]

...Or well, it's not easy for me ^^U hehe

Anyways, here it is.

The circles C: x^2 + y^2 + kx + (1+k)y - (k+1) = 0, pass through the same two points for every real number k

(1) Find the coordinates of these two points
(2) Find the Minimum value of the radius of a circle C

Thanks a lot in advance for your help!!

 

[pka]

The circles C: x^2 + y^2 + kx + (1+k)y - (k+1) = 0, pass through the same two points for every real number k.

Completing the squares we get:
Center at \(\displaystyle \L
\left( {\frac{{ - k}}{2},\frac{{ - \left( {k + 1} \right)}}{2}} \right)\)

and radius \(\displaystyle \L
\frac{{\sqrt {2k^2 + 6k + 5} }}{2}\).

Thus the circle is \(\displaystyle \L
\left( {x + \frac{k}{2}} \right)^2 + \left( {y + \frac{{k + 1}}{2}} \right)^2 = \frac{{2k^2 + 6k + 5}}{4}\).

Choose two values of k and solve.