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?

Options

A. 1
B. 2
C. 3
D. 4

Answer

B

Explanation

Given circle and the coordinate axes will have exactly three common points in the following cases:
`ul("CASE I")` When the circle passes through the origin and does not touch either of the axes
In this, (0, 0) must satisfy the equation `x^(2)+y^(2)+2x+4y-p=0`.
`:. 0-p=0rArr p=0`
`ul("CASE II")` When the circle touches x-axis and intersect y-axis in two distinct points:
In this case, y-coordinates of centre = Radius and , y-intercept `gt0`
`rArr -2=sqrt(p+5) and 2sqrt(4+p) gt 0`
`rArr 4=p+5 and p+4 gt 0`
`rArr p=-1 and p gt - 4`
`rArr p=-1`
`ul("CASE")` When the circle touches y-axis and intersects x-axis in two distinct points:
In this case,
x-coordinates of centre = Radius and, x-intercept `gt0`
`rArr -1=sqrt(p)+5 and 2sqrt(1)+p gt 0`
`rArr p+5=1 and p+1 gt 0`
`rArr p=-4 and p gt - 1`
This is not possible. Hence, there are two values of P.