Question

Show that the lines given by `(x+1)/-10=(y+3)/-1=(z-4)/1`  and `(x+10)/-1=(y+1)/-3=(z-1)/4` intersect. Also find the co-ordinates of the point of intersection.

Answer 1

The given lines are intersecting direction ratio of lines not proportional

`Let (x+1)/-10=(y+3)/-1=(z-4)/1=lambda`

therefore The coordinates of a point on the line are `(-10 lambda-1,-lambda-3,lambda+4)`

Similarly, let, `(x+10)/-1=(y+1)/-3=(z-1)/4=mu`

The coordinates of a point on this line are `(-mu-10,-3mu-1,4mu+1)`

Since two lines intersect for some value of ` lambda and mu.`

`therefore (-10lambda-1,-lambda-3,lambda+4)=(-mu-10,-3mu-1,4mu+1)`

`-10lambda-1=-mu-10 , -lambda-3=-3mu-1, lambda+4=4mu+1`

`-10lambda+mu=-9 .........(1)`

`-lambda+3mu=2 ............(2)`

`lambda-4mu=-3..............(3)`

Solving equation (1) and (2), we get `lambda= I, mu=1` and 3^rd equation holds for these values.
∴ The lines intersect at the point (–11, –4, 5)