Show that the lines :

(x-1) /2 = (y-2) /3 = (z-3) /4 and (x-4) /5 = (y-1) 2 = z

intersect. find their point of intersection.

Question

Show that the lines :
(x-1) /2 = (y-2) /3 = (z-3) /4 and (x-4) /5 = (y-1) 2 = z
intersect find their point of intersection

Ajanta Trivedi · Accepted Answer

the equation of the line is

let $\frac{x - 1}{2} = \frac{y - 2}{3} = \frac{z - 3}{4} = t$ .........(1)

the coordinates of any point on the line is given as:

x=2t+1 , y=3t+2 and z=4t+3.........(1)'

the equation of the another line is

let $\frac{x - 4}{5} = \frac{y - 1}{2} = \frac{z - 0}{1} = s$ ......(2)

the coordinates of any point on the line (2) is given as:

x=5s+4 , y=2s+1 , z=s......(2)'

now if the lines intersect , the intersection point will lie on both the equation.

2t+1=5s+4

$2 t - 5 s = 3........... (3)$

$3 t + 2 = 2 s + 1 3 t - 2 s = - 1......... (4)$

now solving (3) and (4):

3*(3)-2*(4)

$6 t - 15 s = 9 6 t - 4 s = - 2 - 11 s = 11 s = - 1$

from equation ;(3)

2t+5=3

2t=-2

t=-1

now the z-coordinates of equation (1) is 4*(-1)+3=-4+3=-1

z-coordinates of equation (2) is -1

which are equal.

thus the lines intersect and their intersection point is $(5 * (- 1) + 4, 2 * (- 1) + 1, - 1) = (- 1, - 1, - 1)$