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

Answer 1

The centre of the given circle is $(-1,-2)$. For exactly three intercepts with the coordinate axes, there are three possibilities:

1. Tangent to $Y$-axis and intersecting $X$-axis in $2$ points.

2. Tangent to $X$-axis and intersecting $Y$-axis in $2$ points.

3. Passing through origin and intersecting the coordinate axes in one point each.

In this case, if we take the circle tangent to $Y$-axis, it will not intersect the $X$-axis in any point. So, case 1 is dismissed.

Hence, there are $2$ possibilities as shown in the given figure.

Hence, $p$ can take $2$ values only.