圓錐曲線

[happy520]

圓錐曲線

[thepiano]

第 1 題
令橢圓方程式為 x^2/a^2 + y^2/b^2 = 1
那兩點代進去解聯立
可得 a^2 = 16，b^2 = 9


第 2 題
令雙曲線方程式為 (x - 5)^2/a^2 - (y + 2)^2/b^2 = 1
b/a = 3/4
2a = AA' = 6
a = 3，b = 9/4


第 3 題
2x^2 - y^2 = 1
微分
4x - 2yy' = 0
y' = 2x/y

設過 (-2，5) 的切線與 2x^2 - y^2 = 1 相切於 (s，t)
則
2s^2 - t^2 = 1
5 - t = (2s/t)(-2 - s)

解聯立得
s = 1，t = -1
s = -13/7，t = 7/17

所求為 -2 or -26/7

[happy520]

解聯立得
s = 1，t = -1
s = -13/7，t = 7/17

所求為 -2 or -26/7

這裡不懂,可麻煩解釋嗎

[thepiano]

2s^2 - t^2 = 1 ...... (1)
5 - t = (2s/t)(-2 - s) ...... (2)

由 (2) 2s^2 - t^2 = -4s - 5t 代入 (1)
-4s - 5t = 1
s = -(5t + 1) / 4 代入 (1) 整理得
17t^2 + 10t - 7 = 0
t = -1 or 7/17
s = 1 or -13/7

所求之斜率 = 2s/t = -2 or -26/7