Fixed points through a general circle.

Question

The circle $C: x^2 + y^2 + kx + (1+k)y - (k+1)=0$ passes through two fixed points for every real number $k$. Find $(i)$ co-ordinates of these two points and $(ii)$ the minimum value of the radius.

Answer 1

HINT:

This is an arbitrary circle which passes through the intersection of $x^2+y^2+y-1=0$ & $x+y-1=0$

For the second question :

$$x^2+y^2+kx+(1+k)y-(k+1)=0$$ $$\implies \left(x+\frac k2\right)^2+\left(y+\frac{1+k}2\right)^2=k+1+\frac{k^2}4+\frac{(k+1)^2}4$$

If $r$ is the radius, $$r^2=k+1+\frac{k^2}4+\frac{(k+1)^2}4=\frac{4k+4+k^2+(k+1)^2}4=\frac{(2k+3)^2+1}8\ge \frac18$$