Question

Find the the differential equation for all the straight lines, which are at a unit distance from the origin.

Answer 1

The general equation of a line that is at unit distance from the origin is given by

`xcosα+ysinα=1      .....(i)`

Differentiating (i) w.r.t. x, we get

`cosα+dy/dxsinα=0`

`⇒cotα=−dy/dx           .....(ii)`

Dividing (i) by sinα, we get

`x cosα/sinα+ysinα/sinα=1/sinα `

`⇒xcotα+y=cosecα`

`⇒xcotα+y=sqrt(1+cot^2α)      .....(iii)`

 

Putting the value of (ii) in (iii), we get

`x(−dy/dx)+y=sqrt(1+(−dy/dx)^2)  .....(iv)`

Squaring (iv), we get

`(−xdy/dx+y)^2=(sqrt(1+(dy/dx)^2))^2`

`(x^2−1)(dy/dx)^2−2xydy/dx+y^2−1=0`