It is known that the center of the ellipse is at the origin, the focus is on the x-axis, the length of the major axis is equal to 12, and the eccentricity is 13. (1) find the standard equation of the ellipse; (2) make a straight line L through the left vertex of the ellipse. If the distance between the moving point m and the right focus of the ellipse is less than 4, find the trajectory equation of the point M

It is known that the center of the ellipse is at the origin, the focus is on the x-axis, the length of the major axis is equal to 12, and the eccentricity is 13. (1) find the standard equation of the ellipse; (2) make a straight line L through the left vertex of the ellipse. If the distance between the moving point m and the right focus of the ellipse is less than 4, find the trajectory equation of the point M

(1) Let the semi major axis length of the ellipse be a, the semi minor axis length be B, and the semi focal length be c. as we know, 2A = 12, so a = 6. (2 points) and Ca = 13, that is, a = 3C, so 3C = 6, that is, C = 2. (4 points) then B2 = a2-c2 = 36-4 = 32. Because the focus of the ellipse is on the X axis, the standard equation of the ellipse is x236 + y232 = 1. (6 points) (2) method 1: because a = 6, the equation of the straight line L is x = - 6, and C = 2 =2, so the right focus is F2 (2, 0) passing through the point m, making the vertical line of the straight line L, and the perpendicular foot is h. let's set the problem, | MF2 | = | MH | - 4. Let's set the point m (x, y), then (x − 2) 2 + y2 = (x + 6) − 4 = x + 2. (8 points) two sides of the square, and get (X-2) 2 + y2 = (x + 2) 2, that is, y2 = 8x. (10 points). Therefore, the trajectory equation of point m is y2 = 8x. (12 points) method 2: because a = 6, C = 2, so a-c = 4, so the left focus of the ellipse F1 to The distance of the straight line L is 4. (8 points) suppose that the distance from the moving point m to the right focus of the ellipse is equal to the distance from it to the straight line x = - 2, so the trajectory of point m is a parabola with the right focus of F2 (2,0) as the focus and the straight line x = - 2 as the collimator. (10 points) obviously, the vertex of the parabola is at the original coordinate point, and P = | F1F2 | = 4, so the trajectory equation of point m is y2 = 8x. (12 points)