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?

Answer

2

Explanation

Case `I :` When `p=0` (i.e., circle passes through origin ).
Here, equation of circle `x^(2)+y^(2)+2x+4y=0`.
Clearly, this circle meets coordinate axes at exactly three points.
Case `II :` When circle intersects x-axis at 2 distinct points and touches y - axis.
This occurs when `(g^(2)-c) gt 0` and `f^(2)-c=0`
`implies 1- (-p) gt0` and `4- (-p) =0`
`implies p gt -1` and `p = -4`
Not possible .
Case` III :` When circles intersects y-axis at 2 distinct points and touches x-axis.
This occurs when `g^(2)-x=0` and `(f^(2)-c) gt0`
`implies 1- (-p)=0 ` and `4-(-p) gt0`
`implies p = -1` and `implies p gt -4`
Here, `p = -1` is possible
Hence, possible values of p are 0 and -1.