MathDB

Circles with centers in a set

Source: RMO Maharashtra and Goa 2019 P6

November 10, 2019

algebrageometrycoordinate geometryanalytic geometryconicsparabola

Problem Statement

Let

k

be a positive real number. In the

X-Y

coordinate plane, let

S

be the set of all points of the form

(x,x^2+k)

where

x\in\mathbb{R}

. Let

C

be the set of all circles whose center lies in

S

, and which are tangent to

X

-axis. Find the minimum value of

k

such that any two circles in

C

have at least one point of intersection.