Two cars P and Q are moving at uniform speeds, 50 km/hr and 25 km/hr respectively, on two straight roads intersecting at right angle to each other. P passes the intersecting point of the roads when Q has still to travel 50 km to reach it. What is the shortest distance between the cars during the journey?

Lets assume P is going on the X axis to the east and Q is moving north on Y axis.

At time t = 0, P is at the origin (0,0) and Q is at (0, -50).
At time t = T, P is at (50T,0) and Q is at (0,-50 + 25T)

Distance between P and Q is $$ \sqrt {(50T)^{2} + (25T-50)^{2}}$$

= $$ \sqrt {3125T^{2} + 2500 - 2500T}$$
For that to be minimum, $$ 3125T^{2} + 2500 - 2500T $$ is minimum, ie derivative is 0
6250T - 2500 = 0 ie T = .4

Substituting T, we get distance = $$ 20 \sqrt (5)$$