(2) After analysis, when the slope of the straight line M does not exist, it does not satisfy that A is the midpoint of PB. Then the oblique equation of the straight line m is established, combined with the elliptic equation, and then arranged. Write x 1+x2, x 1x2 by using the relationship between roots and coefficients, and get the equation about k by combining 2x 1=x2, so that the slope of the straight line m can be obtained.
Solution: (i) The distance from point M(x, y) to straight line x=4 is twice that from point n (1, 0) to straight line x = 4, then. . . . . .
The trajectory of the moving point m is an ellipse, and the equation is
x2
four
+
y2
three
= 1
(2) p (0 0,3), let A(x 1, y 1) and B(x2, y2), let a be the midpoint of PB, and get 2x 1=0+x2, 2y 1 = 3+y2.
The coordinates of the upper and lower vertices of the ellipse are (0,
three
) and (0,?
three
), the straight line m does not pass through these two points, that is, the slope k of the straight line m exists.
Let the equation of straight line m be: y = kx+3.
. . . . . . . . .
The slope k of the straight line m =+/-.
three
2
.