We have a dynamic circle P that passes through point F₁ at coordinates (2,0) and is externally tangent to the fixed circle F₂ centered at (-2,0) with radius 2. As the dynamic circle moves while maintaining these conditions, its center P traces out a trajectory. Let's visualize this geometric relationship and find the equation of the trajectory.
Let's derive the trajectory equation step by step. Since circle P passes through F₁, the distance from P to F₁ equals the radius of circle P. Since circle P is externally tangent to the fixed circle, the distance from P to F₂ equals the radius plus 2. This gives us the relationship |PF₂| - |PF₁| = 2, which is the definition of a hyperbola with foci at F₁ and F₂.
Now we examine the line l passing through F₁ and intersecting the hyperbola. The correct line is x = 2, which passes through F₁(2,0). When we substitute this into the hyperbola equation, we get the intersection points E(2, 2√3) and F(2, -2√3). We also have point D at (-1, 0), and we draw lines from E and F to point D.
Now we calculate the equations of lines ED and FD, and find where they intersect line l. For line ED with points E(2, 2√3) and D(-1, 0), the slope is 2√3/3. The line equation is y = (2√3/3)(x + 1). When this intersects line l at x = 2, we get point M at (2, 2√3), which coincides with point E. Similarly, line FD intersects l at point N, which coincides with F.