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.

Answer 1

Answer:

The two lines are

$\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{4}=m \\\\ \frac{x-4}{5}=\frac{y-1}{2}=\frac{z-0}{1}=n$

x=2 m +1

y=3 m+2

z=4 m+3

x=5 n+4

y=2 n+1

z=n

Comparing two x, y and z values

1.→2 m+1= 5 n +4

2 m- 5 n= 4-1

2 m -5 n=3-----   ×3

2. →3 m+2=2 n+1

3 m-2 n=1-2

3 m- 2 n= -1-------×2

3.→ 4 m +3=n

6 m - 15 n =9

6 m - 4 n = -2

→6 m- 15 n - 6 m +4 n= 9 +2

-11 n =11

n= -1

Substituting the value of , n in equation 2

3 m - 2×(-1)= -1

3 m +2=-1

3 m= -1 -2

3 m= -3

m=-1

m= -1, and n= -1, satisfy the equation .3

So, The Point of intersection of two lines are

x= -2 +1= -1

y= -3 +2= -1

z= -4 +3= -1

So, point of intersection of two lines are , (-1,-1,-1).

Answer 2

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