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

Solution

## 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.Maths

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