Question

Form the differential equation of family of lines situated at a constant distance `p` from the origin.
A. ` (x^(2)+y^(2))(dy)/(dx)=2y{x-p((dy)/(dx))^(2)}`
B. ` (x(dy)/(dx)-y)^(2)-p^(2){1+((dy)/(dx))^(2)}=0`
C. ` ( x (dy)/(dx)-y)^(2)-(p(dy)/(dx)+x(dy)/(dx))=0`
D. ` (x-y)-((dy)/(dx)-(dx)/(dy))=0`

Answer 1

Correct Answer - b
All lines at a constant distance p from the the origin are tangent to the circle , ` x^(2) + y^(2) = p^(2) rArr ` Equation to the
family of such lines are `y = mx p sqrt(1+m^(2)) " Put m " (dy)/(dx)`
and get the result
i.e `y = x (dy)/(dx) pm p sqrt(1+((dy)/(dx))^(2))`
` rArr (x (dy)/(dx)-y)^(2) = p^(2) [ 1+ ((dy)/(dx))^(2)]`
` rArr ( x (dy)/(dx)-y)^(2) = p^(2) [ 1+((dy)/(dx))^(2)]`