Question

The paths traced by two hot air balloons are: 

`(x − 1)/2 = (y − b)/3 = (z − 3)/4 and (x − 4)/5 = (y − 1)/2 = z/1`

Find the value of ‘b’ to be avoided so that the two hot air balloons do not collide.

Answer 1

Write both lines in terms of constants λ and µ:

x = 2λ + 1, y = 3λ + b, z = 4λ + 3 

x = 5µ + 4, y = 2µ + 1, z = µ

Set the x and z coordinates equal to find the intersection point:

From z: µ = 4λ + 3

From x: 2λ + 1 = 5µ + 4 ⇒ 2λ − 5µ = 3

Substitute µ into the x equation:

2λ − 5(4λ + 3) = 3

2λ − 20λ − 15 = 3

−18λ = 18 ⇒ λ = −1

Then, µ = 4(−1) + 3 ⇒ µ = −1.

Find the value of ‘b’ for the collision.

Set the y coordinates equal and plug in λ = −1, µ = −1

3λ + b = 2µ + 1

3(−1) + b = 2(−1) + 1

−3 + b = −1

b = 2