The paths traced by two hot air balloons are: and Find the value of ' ' to be avoided so that the…

Let coordinates of general points on the two paths be: Line 1: $(2\lambda + 1, 3\lambda + b, 4\lambda + 3)$ Line 2: $(5\mu + 4, 2\mu + 1, \mu)$ For the balloons to collide, the paths must intersect: $2\lambda + 1 = 5\mu + 4 \implies 2\lambda - 5\mu = 3$ $4\lambda + 3 = \mu \implies 4\lambda - \mu = -3$ From the second equation, $\mu = 4\lambda + 3$. Substituting into the first: $2\lambda - 5(4\lambda + 3) = 3 \implies -18\lambda = 18 \implies \lambda = -1$. Then $\mu = 4(-1) + 3 = -1$. Equating $y$-coordinates: $3\lambda + b = 2\mu + 1 \implies 3(-1) + b = 2(-1) + 1 \implies -3 + b = -1 \implies b = 2$. Therefore, the value of $b$ to be avoided so that the balloons do not collide is $b = 2$.