JEE Advanced · 2017 · 21 May 2017 · Paper 1 · Circle

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?

Answer: 2

Solution

Rewrite the circle as

(x + 1)^{2} + (y + 2)^{2} = p + 5

. For exactly three common points with the coordinate axes, either: (1) the circle passes through the origin and is tangent to one axis, or (2) the circle is tangent to both axes and passes through the origin. Case 1: Circle passes through origin:

p + 5 = 1 + 4 = 5

, so

p = 0

. Case 2: Circle tangent to

x

-axis: radius equals distance from center to

x

-axis, so

p + 5 = 2

, giving

p + 5 = 4

and

p = - 1

. Both values give exactly three intersection points. Answer: 2.

JEE Advanced · 2017 · 21 May 2017 · Paper 1 · Integer answer

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