The correct option is 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
Thus the given equation represents a parabola. It may noted that (1,0) is the focus and 4x−3y+1=0 is the directrix of the parabola.