✔ 最佳答案
1)
a)
The required equation of the family of st. lines :
2x + y - 1 + k(3x - 2y - 5) = 0
(2 + 3k)x +(1 - 2k)y - (1 + 5k) = 0 ...... [1]
b)
Find the point of intersection of L(3) and L(4):
L(3): x - 3 = 0 ...... [2]
L(4): x + y - 1 = 0 ...... [3]
From [2]:
x = 3
Put x = 3 into [3]:
(3) + y - 1 = 0
y = -2
To find the required equation, put x = 3 and y = -2 into [1]:
(2 + 3k)(3) + (1 - 2k)(-2) - (1 + 5k) = 0
6 + 9k - 2 + 4k - 1 - 5k = 0
8k + 3 = 0
k = -3/8
The required equation :
[2 + 3(-3/8)]x + [1 - 2(-3/8)]y - [1 + 5(-3/8)] = 0
(7/8)x + (14/8)y + (7/8) = 0
x + 2y + 1 =0
2)
Let m1 be the slope of the required line,
and m2 be the slope of L(3).
The slope of L(3): m2 = 2
|(m1 - m2) / (1 + m1m2)| = tan45°
|(m1 - 2) / (1 + 2m1)| = 1
(m1 - 2) / (1 + 2m1) = ±1
m1 - 2 = 1 + 2m1 ..or.. m1 - 2 = -(1 + 2m1)
m1 = -3 ..or.. m1 = 1/3
The equation for the family of st. line passing through the point ofintersection of L(1) and L(2):
(x - 3y + 2) + k(3x - 4y + 7) = 0
(1 + 3k)x - (3 + 4k)y + (2 + 7k) = 0 ...... [1]
Slope of the required line:
(1 + 3k) / (3 + 4k) = -3 ..or.. (1 + 3k) / (3 + 4k) = 1/3
1 + 3k = -9 - 12k ..or.. 3 + 9k = 3 + 4k
15k = -10 ..or.. 5k = 0
k = -2/3 ..or.. k = 0
Put k = -2/3 into [1]:
[1 + 3(-2/3)]x - [3 + 4(-2/3)]y + [2 + 7(-2/3)] = 0
(-3/3)x - (1/3)y - (8/3) = 0
3x + y + 8 = 0
Put k = 0 into 1:
(1 + 0)x - (3 + 0)y + (2 + 0) = 0
x - 3y + 2 = 0
The required equations:
3x + y+ 8 = 0 ..and.. x - 3y + 2 = 0
2013-10-07 22:21:11 補充:
Check for Q.1 :
The point of intersection of L(1) and L(2): (1, -1)
The point of intersection of L(3) and L(4): (3 - 2)
Equations for the st. line passing through (1,-1) and (3, -2):
x + 2y + 1 = 0
2013-10-07 22:28:43 補充:
Check for Q.2:
The pt. of intersection between L(1) and L(2): (-13/5, -1/5)
The pt. lies on both 3x + y+ 8 = 0 and x - 3y + 2 = 0
For L(3): Slope = 2, angle of inclination = 63.43°
For 3x +y + 8 = 0: Slope = -3, angle of inclination = 108.43°
Angle between the two lines = 108.43° - 63.43° = 45°
2013-10-07 22:32:21 補充:
(Cont'd).....Check for Q.2:
For L(3): Slope = 2, angle of inclination = 63.43°
For x - 3y + 2 = 0: Slope = 1/3, angle of inclination = 18.43°
Angle between the two lines = 63.43° - 18.43° = 45°
2013-10-07 23:45:09 補充:
用 (3x - 4y + 7) + k(x - 3y + 2) = 0 會出現你說的問題,兩邊 k 會約去了。
用 (x - 3y + 2) + k(3x - 4y + 7) = 0 便可以了。
這樣的情況偶發會出現。