Daily questions _191112

Known \ (F. \) Is a parabola \ (C_l: ^ 2 = 2px \ Y) \ ((P> 0) \) focus, \ (E \) circle \ (C_2: (x-4 ) ^ 2 + y ^ 2 = 1 \) at any point, and \ (| EF | \) a maximum value of \ (\ dfrac. 19 {} {}. 4 \) .
\ ((. 1) \) seeking parabola \ (C_l \ ) equation;
\ ((2) \) if \ (M (x_0, y_0) \) \ ((2 \ leqslant y_0 \ leqslant. 4) \) parabolic \ (C_l \) , the through \ (M \) in circle \ (C_2 \) two tangent lines, parabolic cross \ (C_l \) in \ (a, B \) , seeking \ (AB \) in the range of the vertical midpoint coordinates.
Analysis:
\ ((1 ) \) is easy to know a title \ (F. \) coordinates of \ (\ left (\ dfrac {2} {P}, 0 \ right) \) , then\ (| EF | \) a maximum value of \ [\ left | 4- \ dfrac {p} {2} \ right |. + 1 = \ dfrac {19} {4} \] Solutions have \ (p = \ dfrac {1} {2} \ ) or \ (P = \ dfrac {31 is} {2} \) . Therefore, the parabolic equation required to \ (y ^ 2 = x \ ) or \ (y ^ 2 = 31x \ ) .
\ ((2) \) from the title, provided \ [M (2pt_0 ^ 2,2pt_0) , A (2pt_1 ^ 2,2pt_1), B (2pt_2 ^ 2,2pt_2). \] readily available linear \ (MA \ ) the general equation for the \ [x- \ left (t_0 + t_1 \ right) y + 2pt_0t_1 = 0. \] Since the straight line \ (MA \) circle \ (C_2 \) tangent, the center \ (C_2 \) a straight line \ (MA \) a distance \ (1 \) , that is \ [\ left (4 + 2pt_0t_1 \ right) ^ 2 = 1 + \ left (t_0 + t_1 \ right) ^ 2. \] Similarly by \ (MB \) and \ (C_2 \) tangent available\ [\ Left (4 + 2pt_0t_2 \ right) ^ 2 = 1 + \ left (t_0 + t_2 \ right) ^ 2. \] Two formulas for the difference may be too \ [\ left [8 + 2pt_0 \ left (t_1 + t_2 \ right) \ right] \ cdot \ left (t_1-t_2 \ right) \ cdot 2pt_0 = \ left (2t_0 + t_1 + t_2 \ right) \ cdot \ left (t_1-t_2 \ right). \] obviously \ (T_l -t_2 \ NEQ 0 \) , if the note \ (AB \) midpoint abscissa \ (m \) , then \ [m = \ dfrac {2pt_1 + 2pt_2} {2} = \ dfrac {16p ^ 2t_0-2pt_0 } {1-4p ^ 2t_0 ^ 2} = \ dfrac {(8p-1) y_0} {1-y_0 ^ 2}. \] whether \ (p = \ dfrac {1 } {2} \) or \ (\ 31 is {2} {dfrac} \) , \ (m \) are about \ (y_0 \) monotonically increasing function, so \ (m \) is in the range of \ [\ left [- \ dfrac {2 ( 8p-1)} {3}
, - \ dfrac {4 (8p-1)} {15} \ right] \]. the case when a \ (P = \ dfrac12 \) , \ (AB \)The ordinate is the midpoint of the range \ (\ left [-2, - \. 4 {dfrac. 5} {} \ right] \) .
Case 2 When \ (P = \ dfrac 31 is {2} {} \) , \ (AB \) ordinate is the midpoint of the range \ (\ left [-82, - \. 5 dfrac {} {} 164 is \ right] \) .