高二數學關於拋物線

1.
抛物線： y² = 4x ...... [1]

抛物線的焦點 F = (1, 0)

PF 的方程式：
(y - 0)/(x - 1) = (2k - 0)/(k² - 1)
2k(x - 1) = (k² - 1)y
2kx - 2k = k²y - y
4kx = 2k²y - 2y + 4k
4x = (2k²y - 2y + 4k)/k ...... [2]

[1] = [2]:
y² = (2k²y - 2y + 4k)/k
ky² + (2 - 2k²)y - 4k = 0
(y - 2k)(ky + 2) = 0
y = 2k 或 y = -2/k

將 y = -2/k 代入 [1] 中：
(-2/k)² = 4x
x = 1/k²

Q 的坐標 = (1/k², -2/k)

ΔPQR 面積
__|3_____0|
=_|k²____2k| x (1/2)
__|1/k²_-2/k|
__|3_____0|
= (1/2) x [6k - 2k - 2/k + 6/k]
= 2k + (2/k)


(b)
抛物線 y = kx² ...... [1]
O 的坐標 = (0, 0)

設 OA 的斜率為 m (m >0)。
則 OB 的斜率為 -1/m。

OA : y = mx ...... [2]

[1] = [2] :
kx² = mx
kx² - mx = 0
x(kx - m) = 0
x = 0 或 x = m/k

將 x = m/k 代入 [2] 中：
y = m(m/k)
y = m²/k

A 的坐標 = (m/k, m²/k)

OB : y = -x/m ...... [3]

[1] = [3] :
kx² = -x/m
kx² + (x/m) = 0
kmx² + x = 0
x(kmx + 1) = 0
x = 0 或 x = -1/mk

將 x = -1/mk 代入 [3] 中：
y = -(-1/mk)/m
y = 1/m²k

B 的坐標 = (-1/mk, 1/m²k)

ΔOAB 面積
__|0______-__0|
=_|m/k__-__m²/k| x (1/2)
__|-1/mk__1/m²k|
__|0___-_____0|
= (1/2) x [(1/mk²) + (m/k²)]
= (1/2) x (1 + m²) / mk²
= [(1/2) x (1 - 2m + m²) / mk²] + [(1/2) x 2m / mk²]
= [(1/2) x [(1 - m)² / mk²] + (1/k²)

因為已設 m > 0，所以 [(1/2) x [(1 - m)² / mk²] > 0
故此，ΔOAB 面積 = [(1/2) x [(1 - m)² / mk²] + (1/k²) > (1/k²)
ΔOAB 的最小面積 = 1/k²