拋物線內一點A到拋物線上一點與焦點距離和的最小值
Author: 藍邦偉  老師

A

P

F

$ 拋物線與距離 $

$ 已知A點為拋物線內一點，F$
$為拋物線的焦點，P點在拋物$
$線上，求\overline{PA}+\overline{PF}的最小值 $

H

1

$ 顯示正解 $

2

$ 顯示輔助物件 $

u = Vector((x(Corner[1]), 0), (x(Corner[3]), 0))
v = Vector((0, y(Corner[1])), (0, y(Corner[3])))
O = (0, 0)
Q_2: 0.7 * Corner[1] + 0.3 * Corner[2]
Q_3: (x(Q_2), y(Corner[4]))
poly1, q_1, q_2, q_4, q_3 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
c = 0.9 (slider from 0.1 to 3, step 0.1)
F: (-2 + 2 * c, 0)
V: (-2 + c, 0)
Γ: (y - y(V))^(2) = 4 * c * (x - x(V))
A = (3.1071716868740102, 1.531408741270718)
P = Point(Γ)
b = Segment(P, A)
d = Segment(P, F)
H: (-2, y(P))
e = Segment(P, H)
text3: "     $ \overline{PA}+\overline{PF}$
$=" + (LaTeX[b]) + "+" + (LaTeX[d]) + "$
$=" + (LaTeX[b + d]) + " $"
f = false
g = false
text4: "     $ \overline{PA}+\overline{PF} $
$=\overline{PA}+d(P,L)有最小值$
$⇒d(A,L)=" + (LaTeX[x(A) - x(H)]) + "$"
Q: (-2, y(A))
h = Segment(A, Q)
B = Intersect(Γ, h)