The equation of family of curves for which the length of normal is equal to the radius vector

Dear pallvi

let a general point on the curve (x,y)

 length of radius vector = √x² +y²

 lengt of normal is  =y√(1+(dy/dx)²)

given   √x² +y²   =y√(1+(dy/dx)²)

   or   x² +y²   =y²(1+(dy/dx)²)

 or     x²    =y²(dy/dx)²

 or  dy/dx = ±x/y

    ydy  =±xdx

    y²/2  =± x²/2  +c

   y²  ± x²  =c

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