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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.

682617_640542_ans_22639192018149999ff50f90fd80fb2c.png

Maths

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