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?

Explanation

To determine how many values of \( p \) allow the circle given by the equation \( x^2 + y^2 + 2x + 4y - p = 0 \) to intersect the coordinate axes at exactly three points, we will analyze the conditions under which this occurs.

### Step 1: Rewrite the Circle Equation
The equation of the circle can be rewritten in standard form by completing the square.

1. **Complete the square for \( x \) and \( y \)**:
\[ x^2 + 2x + y^2 + 4y - p = 0 \]
Completing the square for \( x \):
\[ x^2 + 2x = (x+1)^2 - 1 \]
Completing the square for \( y \):
\[ y^2 + 4y = (y+2)^2 - 4 \]
Substituting these back into the equation gives:
\[ (x+1)^2 + (y+2)^2 - 5 - p = 0 \]
Thus, the equation becomes:
\[ (x+1)^2 + (y+2)^2 = p + 5 \]
This represents a circle centered at \( (-1, -2) \) with radius \( \sqrt{p + 5} \).

### Step 2: Analyze Intersection with the Axes
To find the conditions for the circle to intersect the coordinate axes at exactly three points, we consider different cases:

#### Case 1: Circle passes through the origin
- If \( p = 0 \), the equation becomes:
\[ (x+1)^2 + (y+2)^2 = 5 \]
The circle passes through the origin (0, 0), and it intersects the axes at three points: the origin and two more points on the axes.

#### Case 2: Circle intersects the x-axis at two distinct points and touches the y-axis
- The conditions for this case are:
- The circle intersects the x-axis at two points: \( g^2 - c > 0 \)
- The circle touches the y-axis: \( f^2 - c = 0 \)

Here, \( g = 1 \), \( f = 4 \), and \( c = -p \):
- From \( g^2 - c > 0 \):
\[ 1^2 - (-p) > 0 \implies 1 + p > 0 \implies p > -1 \]
- From \( f^2 - c = 0 \):
\[ 4^2 - (-p) = 0 \implies 16 + p = 0 \implies p = -16 \]
This case does not yield valid values since \( p = -16 \) does not satisfy \( p > -1 \).

#### Case 3: Circle intersects the y-axis at two distinct points and touches the x-axis
- The conditions for this case are:
- The circle touches the x-axis: \( g^2 - c = 0 \)
- The circle intersects the y-axis at two points: \( f^2 - c > 0 \)

Using the same definitions:
- From \( g^2 - c = 0 \):
\[ 1^2 - (-p) = 0 \implies 1 + p = 0 \implies p = -1 \]
- From \( f^2 - c > 0 \):
\[ 4^2 - (-p) > 0 \implies 16 + p > 0 \implies p > -16 \]
Here, \( p = -1 \) satisfies \( p > -16 \).

### Conclusion
From the analysis, we find two possible values for \( p \):
1. \( p = 0 \) (circle passes through the origin)
2. \( p = -1 \) (circle touches the x-axis and intersects the y-axis at two points)

Thus, the total number of values of \( p \) for which the circle and the coordinate axes have exactly three common points is **2**.