F.4 - A.Maths - 圓與直線之交點

1. C: x2 +y2 = 5 ─── (1)
L: 2y = x + 5, x = 2y- 5 ─── (2)
代(2)入(1):
(2y - 5)2+ y2 = 5
4y2 – 20y+ 25 + y2 = 5
y2 – 4y +4 = 0
(y - 2)2 =0
y = 2
x = 2(2) – 5 = -1
交點(-1 , 2)

3. C: x2 + y2+ 10x + 5y + 25 = 0 ─── (1)
L: x = 3y ─── (2)
代(2)入(1):
(3y)2 + y2+ 10(3y) + 5y + 25 = 0
10y2 + 35y +25 = 0
2y2 + 7y + 5 =0
(2y + 5)(y + 1) = 0
y = -1 或 -5/2
當y = -1, x = 3(-1) = -3
當y = -5/2, x = 3(-5/2) = -15/2
交點(-3 , -1)或(-15/2 , -5/2)

4. C: x2 + y2– 6x – 4y + 12 = 0 ─── (1)
L: x + y = 6, x = -y +6 ─── (2)
代(2)入(1):
(-y + 6)2 +y2 – 6(-y + 6) – 4y + 12 = 0
y2 – 12y +36 + y2 + 6y – 36 – 4y + 12 = 0
2y2 – 10y +12 = 0
y2 – 5y + 6= 0
(y - 3)(y - 2) = 0
y = 2 或 3
當y = 2, x = -2 + 6 = 4
當y = 3, x = -3 + 6 = 3
交點(3 , 3), (4 , 2)