Question

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? .
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Answer 1

$x^2+y^2+2x+4y-p=0g=-1,f=-2$
There are three such possible cases:
Case 1: When circle passes through origin
$\Rightarrow p=0$

Case 2: When circle touches y-axis and cuts x-axis
$f^2=c\&g^2>c\Rightarrow p=-4\&1>-p\Rightarrow 1+p>0\Rightarrow \text{Not possible}$

Case 3: When circle touches x-axis and cuts y-axis
$g^2=c\&f^2>c\Rightarrow p=-1\&4>-p\Rightarrow 4+p>0\therefore \text{Only two possible value of}p\text{i.e.,}p=0,-1$