For how many values of p, the circle $x^{2}+y^{2}+2x+4y-p=0$ and the coordinate axes have exactly three common points?

For how many values of p, the circle x²+y²+2x+4y-p=0 and the coordinate axes have exactly three common points?

$S: x^2+y^2+2x+4y-p=0$

$\text{axes: } x=0 \ (\text{y-axis}),\ y=0 \ (\text{x-axis})$

$\text{circle } \cap \text{ axes has exactly } 3 \text{ points}$

$\text{To find: number of values of } p$

$x^2+y^2+2x+4y-p=0$ $\Rightarrow (x+1)^2+(y+2)^2 = p+5$ $\Rightarrow C(-1,-2),\quad r=\sqrt{p+5}$ $d(C,\ y\text{-axis}) = |x_C| = 1$ $d(C,\ x\text{-axis}) = |y_C| = 2$ $1.\ \text{centre and radius}$ $2.\ \text{intersections with each axis}$ $3.\ \text{count total, find when } N = 3$ $x=0:\ y^2+4y-p=0$ $\Delta = 16+4p = 4(p+4)$ $N_y = 2\ (p>-4),\ 1\ (p=-4),\ 0\ (p<-4)$ $y=0:\ x^2+2x-p=0$ $\Delta = 4+4p = 4(p+1)$ $N_x = 2\ (p>-1),\ 1\ (p=-1),\ 0\ (p<-1)$ $N = N_y + N_x - \delta$ $\delta = 1 \text{ if } (0,0) \text{ on circle, else } 0$ $(0,0):\ -p = 0 \Rightarrow p=0$ $p>-1:\ N_y=2,\ N_x=2 \Rightarrow N=4$ $p=0:\ (0,0) \text{ on circle} \Rightarrow N=4-1=3 \ \checkmark$ $p=-1:\ r=2=d(C,x\text{-axis})$ $\Rightarrow N_y=2,\ N_x=1 \Rightarrow N=3 \ \checkmark$ $-4<p<-1:\ N_y=2,\ N_x=0 \Rightarrow N=2$ $p=-4:\ r=1=d(C,y\text{-axis}) \Rightarrow N=1$ $p<-4\ (\text{and } p>-5):\ N=0$ $N=3 \text{ only for } p=0 \text{ and } p=-1$ $\boxed{p \in \{-1,\ 0\} \Rightarrow 2 \text{ values}}$