A curve that passes through (2, 4) and having subnormal of constant length of 8 units can be?

Dear Rishi,

Let the curve be y = f(x). Subnormal at any point = |y*dy/dx|

                                 y*dy/dx = ±8  ;which indicates y dy = ±8dx ;which indicates  = ±8x + c

                                 which indicates: y² = 16 x+2c₁, 

                                                        : c₁= -8    or  y² = -16x +2c₂,  

                                                         :c₂= 24

                              Hence :        y² = 16x – 8; y² = -16x + 24  ;

would be the desired equations.

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length of subnormal =y(dy/dx)

=8 ......(given)

y(dy/dx)=8

y dy = 8 dx

integrating on both sides

y^2 =16x +c (c is integrating constant)

satisfying by (2,4)

we get y^2 = 16(x-1)

this is the equation of required curve.

From,

       PRAVIN ULLAGADDI.