✔ 最佳答案
1. x^4 - x^2y = 1
對兩邊以x作微分
4x^3 - x^2 dy/dx - 2xy = 0
dy/dx = (4x^2 - 2y) / x
當曲線與x軸相交，y座標 = 0
所以，x^4 = 1, x = 1 或 -1
當x = 1，切線的斜率 = [4(1)^2 - 0] / 1 = 4
法線的斜率 = -1/4
切線方程：(y - 0) / (x - 1) = 4, 4x - y - 4 = 0
法線方程：(y - 0) / (x - 1) = -1/4, x + 4y - 1 = 0
當x = -1，切線的斜率 = [4(-1)^2 - 0] / -1 = -4
法線的斜率 = 1/4
切線方程：(y - 0) / [x - (-1)] = -4, 4x + y + 4 = 0
法線方程：(y - 0) / [x - (-1)] = 1/4, x - 4y + 1 = 0
2. 9x - 8y - 2 = 0 的斜率 = 9 / 8
9x^2 + 16y^2 = 52
對兩邊以x作微分
18x + 32y dy/dx = 0
dy/dx = -9x / 16y
所以，-9x / 16y = 9 / 8
x = -2y
代入曲線方程：9(-2y)^2 + 16y^2 = 52
y = 1 或 -1
當y = 1，x = -2，切線方程：(y - 1) / [x - (-2)] = 9 / 8
9x - 8y + 26 = 0
當y = -1，x = 2，切線方程：[y - (-1)] / (x - 2) = 9 / 8
9x - 8y - 26 = 0
3. x + 3y + 4 = 0，斜率 = -1/3
所以，所求切線斜率 = 3
y = x^3 + 5
對兩邊以x作微分
dy/dx = 3x^2
所以，3x^2 = 3，x = 1 或 -1
當x = 1，y = 6，切線方程：(y - 6) / (x - 1) = 3，3x - y + 3 = 0
當x = -1，y = 4，切線方程：(y - 4) / [x - (-1)] = 3，3x - y + 7 = 0
2009-03-04 17:08:19 補充：
4. y = x^2 + 7x + 2
dy/dx = 2x + 7
切線斜率 = tan135* = -1
所以，2x + 7 = -1, x = -4, 因此y = (-4)^2 + 7(-4) + 2 = -10
所以，切線方程：
[y - (-10)] / [x - (-4)] = -1
y + 10 = -x - 4
x + y + 14 = 0
