Given the parabola y & #178; = 2x, set point a (a, 0) (a > 0), find the coordinates of point P nearest to point a on the parabola and the corresponding distance | PA |

Given the parabola y & #178; = 2x, set point a (a, 0) (a > 0), find the coordinates of point P nearest to point a on the parabola and the corresponding distance | PA |

The square of distance formula PA & # 178; = (x-a) & # 178; + (y-0) & # 178; = (x-a) & # 178; + Y & # 178; = (x-a) & # 178; + 2x (x > = 0) PA & # 178; = x & # 178; - 2 (A-1) x + A & # 178; = [x - (A-1)] & # 178; + A & # 178; - (A-1) & # 178;
When a > = 1, the minimum value of A-1 > = 0 is obtained at the vertex, PA & # 178; = A & # 178; - (A-1) &# 178; = 2a-1, then x = A-1 P (A-1, ± radical (2a-2)) Pa = radical (2a-1))
When 0

Given the parabola y = x2-kx-k-1, the value of K is obtained according to the following conditions
(1) The vertex is on the X axis;
(2) The vertex is on the Y axis;
(3) The parabola passes through the origin;
(4) The minimum value is - 1

The vertex of y = x2-kx-k-1 is [K / 2, (- 4k-4-k ^ 2) / 4]
1. When (- 4k-4-k ^ 2) / 4 = 0, k = - 4, the vertex is on the X axis
2. When K / 2 = 0 and K = 0, the vertex is on the y-axis;
3. When - k-1 = 0, k = - 1, the parabola passes through the origin
4. When (- 4k-4-k ^ 2) / 4 = - 1, k = 0 or K = - 4, the minimum value is - 1

F is the focal point of the parabola y ^ 2 = = 2px, a (4,2) is the internal fixed point of the parabola, P is a point on the parabola, and the minimum value of PA + PF is 8

PF is the distance from P to the guide line x = - P / 2
PA + PF is the minimum when p is on the line perpendicular to the guide line through a
The minimum value is 4 + P / 2 = 8
p＝8
Equation: y ^ 2 = 16x