Question

Find the shortest distance between the lines x minus 1 by 2 is equal to y - 2 by 3 is equal to z - 3 by 4

Answer 1

Question: Find the shortest distance between the straight lines

(x - 1)/2 = (y - 2)/3 = (z - 3)/4 &

(x - 5)/4 = (y - 4)/4 = (z - 5)/5 .

Solution:

Let P and Q be two points on the straight lines such that PQ is the shortest distance. Let the co-ordinates of P and Q are

(2r + 1, 3r + 2, 4r + 3) & (4r' + 5, 4r' + 4, 5r' + 5)

respectively.

Then the d.r.'s of PQ be

2r - 4r' - 4, 3r - 4r' - 2, 4r - 5r' - 2

Since PQ is perpendicular to both the given straight lines,

4r - 8r' - 8 + 9r - 12r' - 6 + 16r - 20r' - 8 = 0 &

8r - 16r' - 16 + 12r - 16r' - 8 + 20r - 25r' - 10 = 0

i.e., 29r - 40r' - 22 = 0 ..... (1)

40r - 57r' - 34 = 0 ..... (2)

Solving from (1) and (2), we get

r = - 2, r' = - 2

Thus the two points are (- 3, - 4, - 5) & (- 3, - 4, - 5)

Therefore PQ, the shortest distance is 0.