The correct option is B a parabola
The given equation is
25(x2+y2−2x+1)=(4x−3y+1)2 ⋯(1)
The right hand side of this equation can be written in the form of perpendicular distance
(4x−3y+1)2=25⎛⎜
⎜⎝4x−3y+1√(42+32)⎞⎟
⎟⎠2
Then equation (1) can be re-written as
25[(x−1)2+(y−0)2]=25⎛⎜
⎜⎝4x−3y+1√(42+32)⎞⎟
⎟⎠2
⇒√(x−1)2+(y−0)2=|4x−3y+1|√42+32
Here, e=1
The fixed point (1,0) does not lie on the fixed line 4x−3y+1=0
Thus, the given equation represents a parabola.