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?

To determine how many values of \( p \) allow the circle defined by the equation \( x^2 + y^2 + 2x + 4y - p = 0 \) to intersect the coordinate axes at exactly three points, we need to analyze the conditions under which this occurs. The coordinate axes consist of the x-axis (where \( y = 0 \)) and the y-axis (where \( x = 0 \)). Let's break this down step by step.

Rearranging the Circle's Equation

The equation of the circle can be rewritten in standard form. We start by completing the square for \( x \) and \( y \).

• For \( x \): \( x^2 + 2x \) can be rewritten as \( (x + 1)^2 - 1 \).
• For \( y \): \( y^2 + 4y \) can be rewritten as \( (y + 2)^2 - 4 \).

Substituting these into the original equation gives us:

\((x + 1)^2 - 1 + (y + 2)^2 - 4 - p = 0\)

This simplifies to:

\((x + 1)^2 + (y + 2)^2 = p + 5\

Circle's Center and Radius

The circle is centered at \( (-1, -2) \) with a radius of \( \sqrt{p + 5} \). For the circle to intersect the coordinate axes exactly three times, we need to analyze its intersection with each axis.

Intersections with the Axes

1. **Intersection with the x-axis**:

Set \( y = 0 \) in the circle's equation:

\((x + 1)^2 + (0 + 2)^2 = p + 5\

This simplifies to:

\((x + 1)^2 + 4 = p + 5\)

Therefore, \((x + 1)^2 = p + 1\). The points of intersection will yield:

• Two points if \( p + 1 > 0 \) (i.e., \( p > -1 \))
• One point if \( p + 1 = 0 \) (i.e., \( p = -1 \))

2. **Intersection with the y-axis**:

Set \( x = 0 \) in the circle's equation:

\((0 + 1)^2 + (y + 2)^2 = p + 5\

This simplifies to:

\(1 + (y + 2)^2 = p + 5\)

Hence, \((y + 2)^2 = p + 4\). The points of intersection will yield:

• Two points if \( p + 4 > 0 \) (i.e., \( p > -4 \))
• One point if \( p + 4 = 0 \) (i.e., \( p = -4 \))

Finding the Values of p

Now, for the circle to intersect the axes at exactly three points:

• It must intersect the x-axis at 2 points (which occurs when \( p > -1 \))
• It must intersect the y-axis at 1 point (which occurs when \( p = -4 \))

Therefore, the only value of \( p \) that satisfies these conditions is \( p = -4 \). This means that for this value of \( p \), the circle will intersect the x-axis at two distinct points and the y-axis at exactly one point.

Final Result

In conclusion, the only value of \( p \) that allows the circle to intersect the coordinate axes at exactly three common points is:

-4

This indicates that there is exactly one such value for \( p \).