Step 1

The angle between the two roads is given as

$ \theta = 45^{\circ}$

Hence we can take the ratios as

$h=k$

$x=k\: and \: y=\sqrt2k$

According to pythagoras theorem we get,

$h^2+x^2=y^2$

Differentiating w.r.t $t$ we get

$2h.\large\frac{dh}{dt}+2x\large\frac{dx}{dt}=2y\large\frac{dy}{dt}$

$ \Rightarrow h.\large\frac{dh}{dt}+x.\large\frac{dx}{dt}=y\large\frac{dy}{dt}$

Step 2

Now substituting the values for $ h, x, \: and \: y$ and $ \large\frac{dx}{dt}=\large\frac{dy}{dt}=v$

$ k.\large\frac{dh}{dt}+k.v=\sqrt2k.v$

$ \Rightarrow \large\frac{dh}{dt}=\sqrt 2v-v$

$ = v ( \sqrt2-1)$

Hence the rate at which they seperate is

$v(\sqrt2-1)$