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

Step-by-step solution

We shall consider 3 cases. Case I: When p = 0 (i.e., circle passes through origin) Now, equation of circle becomes x 2 + y 2 + 2 x + 4 y = 0 Case II: When circle intersect x-axis at 2 distinct points and touches y-axis. g 2 − c > 0 and f 2 = c 1 + p > 0 and p = - 4 so, this case is not possible Case III: When circle intersect y-axis at 2 distinct points and touches x-axis Now, g 2 - c = 0 and f 2 - c > 0 ⇒ 1 - - p = 0 and 4 - - p > 0 ⇒ p = - 1 and, p > - 4 ∴ p = - 1 is possible. ∴ Finally, we conclude that p = 0 ,