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 $p$.