If the moving circle with radius 1 is known to be tangent to the circle (X-5) 2 + (y + 7) 2 = 16, then the trajectory equation of the center of the moving circle is______ ．

If the moving circle with radius 1 is known to be tangent to the circle (X-5) 2 + (y + 7) 2 = 16, then the trajectory equation of the center of the moving circle is______ ．

The center of the circle (X-5) 2 + (y + 7) 2 = 16 is C (5, - 7), and the radius r = 4. ∵ the moving circle with radius 1 is tangent to the circle (X-5) 2 + (y + 7) 2 = 16. When the two circles are inscribed, the distance from the center a of the moving circle to the point C is equal to the absolute value of the difference between the radii of the two circles, | BC | = 4-1 = 3

Given that the moving circle P is inscribed with F1: x ^ 2 + (y + 2) ^ 2 = 121 / 4, and is circumscribed with F2: x ^ 2 + (Y-2) ^ 2 = 1, 4, note that the locus of point P at the center of the moving circle is e
(1) The equation for finding trajectory e
(2) If the line L passes through point F2 and intersects with track e at P and Q points
(i) If the inscribed circle radius of △ f1pq is r = 10 / 9, calculate the area of △ f1pq
(II) set the point m (0, m), ask: is there a real number m, so that no matter how the line L rotates around the point F2, there is MP vector multiplied by MQ vector = 0? If so, find the value of real number m; if not, please explain the reason
(2) Yes, there are two questions
If good, 50 points for you!

1: Because P is inscribed with F1, let the radius of circle p be r, so Pf1 = 11 / 2 - R ①
And P is circumscribed with F2, so PF2 = 1 / 2 + R ②
① So the equation of trajectory e is (y ^ 2) / 9 + (x ^ 2) / 5 = 1