perpendicular distance

Question

The y-intercepts of 2 parallel lines L and R are -1 and -3 respectively and the x-intercept of L is 3. P is a moving point in the rectangular coordinate plane such that the perpendicular distance from P to L is equal to the perpendicular distance from P to R. Denote the locus of P by T.

Describe the geometric relationship between T and L.


甚麼是perpendicular distance??

Pak Wai · Accepted Answer

perpendicular distance = 垂直距離
其實就是字面的意思
(同時,垂直距離是一點一線之間/兩線之間的最短距離)

L:
x/3 + y/(-1) = 1 (intercept form)
x - 3y - 3 = 0

R:
y = (1/3)x - 3 (slope-intercept form)
3y = x - 9
x - 3y - 9 = 0

Let P = (x1, y1):
PL = PR
|(x1 - 3y1 - 3)|/[1^2 + (-3)^2]^(1/2) = |(x1 - 3y1 - 9)|/[1^2 + (-3)^2]^(1/2)
|x1 - 3y1 - 3| = |x1 - 3y1 - 9|
x1 - 3y1 - 3 = -(x1 - 3y1 - 9) or x1 -3y1 - 3 = x1 - 3y1 - 9 (rejected)
2x1 - 6y1 + 6 =  0
x1 - 3y1 + 6 = 0

i.e.
T: x - 3y + 6 = 0
slope of L = 1/3
slope of T = 1/3 = slope of L
-> T and L are parallel.

2012-11-19 08:07:50 補充：
以上面的題目為例

你可以幻想一下, 一條直線, 從 P點開始, 垂直於直線 L而與L相交於Q
那 垂直距離就是 點P和點Q之間的距離

perpendicular distance

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