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