[Math] For how many values of $p$, does the following circle cuts coordinate axis thrice

geometry

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 ?
I worked it out as follows:
Circle is centered at $(-1,-2)$.
Case 1: When circle passes through origin, $p = 0$
Case 2: When circle has $Y$ axis as tangent at $(0,-2)$, $p = 1$
Is this correct?

Best Answer

Your solution is incorrect.Although the answer is correct, I.e. 2 such values of $p$ exist.

Your Case 1 is correct, but for Case 2 : the circle can't be tangent to y-axis and cut x-axis at 2 points simultaneously. (Since for touching y-axis it must have radius $=1$ and it's distance from x-axis is $2$, therefore it won't intersect with x-axis.)

Case 2 must be when the circle is tangent to x-axis at $(-1,0)$ and cuts y-axis at two distinct points.

Yielding $2$ such values of $p$. In fact this is a question from Indian Institute of Technology - Joint Entrance Examination - Advanced, 2017 Paper 1.

(Abbreviated as IIT-JEE Advanced )

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