# Find the length and the equations of the line of shortest distance between the lines given by: (x-3)/1=(y-5)/(-2)=(z-7)/1 and (x+1)/7=(y+1)/(-6)=(z+1)/1 The answer given in the book is:2?(29 )units,(x-1)/2=(y-2)/3=(z-3)/4 There is no typing mistake at all.

### Asked by Manoj | 23rd May, 2013, 05:48: PM

let P and Q be the points on the given lines, respectively. then the general coordinates of P and Q are:

P(k+3, -2k+5, k+7) and Q (7m-1, -6m-1, m-1)

therefore the direction ratios of PQ are (7m-k-4,-6m+2k-6, m-k-8)

now PQ will be the shortest distance if it is perpendicular to both the given lines, therefore by the condition of perpendicularity,

1(7m-k-4) -2(-6m+2k-6) + 1(m-k-8) = 0 (1)

7(7m-k-4) -6(-6m+2k-6) + 1(m-k-8) = 0 (1)

now solving (1) and (2),

m=0 and k = 0

hence the points are

P(3,5,7) and Q (-1,-1,-1)

therefore the shortest distance between the lines

PQ = sqrt((3+1)^2+(5+1)^2 +(7+1)^2) = sqrt(16+36+64) = sqrt(116) = 2sqrt(29)

since we have the points P and Q, the equation of the line which passes through two given points is:

(x-3)/(3+1) = (y-5)/(5+1) = (z-7)/(7+1)

(x-3)/4 = (y-5)/6 = (z-7)/8

(x-3)/2 = (y-5)/3 = (z-7)/4

However,the answer is given as (x-1)/2=(y-2)/3=(z-3)/4 i.e. the line passes through 1,2,3.

The above equation also passes through 1,2 4

So, they are the same equations.

### Answered by | 25th May, 2013, 11:00: PM

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