Question

For how many values of p does the circle $x^2+y^2+2x+4y-p=0$ and the coordinate axes have exactly three common points?
___

Answer 1

The circle and coordinate axes can have 3 common points, if it passes through origin. [P=0]
If circle is cutting one axis and touching other axis.
Only possibility is of touching X-axis and cutting Y-axis [p=-1], because in case when circle touches Y-axis, it doesn't touch X-axis at all as per radius calculation.