This problem beautifully integrates concepts from coordinate geometry, specifically the properties of parabolas and circles. The first critical step involves finding the shortest distance from an external point to a curve, a common application that often involves calculus or the geometric property of normals.

1. Key Concept: Shortest Distance from a Point to a Curve

The shortest distance from an external point to a curve occurs along the normal drawn from that point to the curve. This is because the normal is perpendicular to the tangent at the point of contact. The distance vector from the external point to the point of contact on the curve will be perpendicular to the curve's direction at that point, which is a fundamental condition for minimum distance.

For a parabola $y^2=4px$, the general equation of a normal at a