The differential equation for all the straight lines which are at a unit distance from the origin is:

Since the equation of lines whose distance from origin is unit, is given by $x\cos \alpha +y\sin \alpha =1$    …(i)

Differentiate w.r.t. x, we get $\cos \alpha +\frac{dy}{dx}\sin \alpha =0$  …(ii)

On eliminating the $\text{'}a\text{'}$ with the help of (i) and (ii)

i.e., (i) -x X (ii)

$\Rightarrow \sin \alpha \left(y-x\frac{dy}{dx}\right)=1\Rightarrow \left(y-x\frac{dy}{dx}\right)=\mathrm{cosec}\alpha$    …(iii)

Also $\text{ (ii) }\Rightarrow \frac{dy}{dx}=-\cot \alpha \Rightarrow {\left(\frac{dy}{dx}\right)}^2={\cot }^2\alpha$      …(iv)

Therefore by (iii) and (iv),

$1+{\left(\frac{dy}{dx}\right)}^2={\left(y-x\frac{dy}{dx}\right)}^2$