The correct option is
B equation of the parabola is
(x−y)2=4(x+y−1)Let S be the focus and A be the vertex of the parabola. let K be the point of intersection of the axis and directrix.
since axis is a line passing through S(1,1) and perpendicular to x+y=1.
So, let the equation of the axis be,
x−y+λ=0
This will pass through (1,1) if,
1−1+λ=0⇒λ=0
So, the equation of the axis is x−y=0
The vertex A is the point of intersection of x−y=0 and x+y=1.
Solving these two equations, we get
x=12 and y=12
So, the coordinates of the vertex A are (12,12).
let (x1,y1) be the co-ordinates of k. Then,
⇒ x1+12=12,y1+12=12
⇒ x1=0, y1=0
So, the co-ordinates of K are (0,0).
Since directrix is a line passing through K(0,0) and parallel to x+y=1.
∴ Equation of the directrix is
y−0=−1(x−0), i.e.x+y=0
Let P(x,y) be any point on the parabola.
Then, distance of P from the focus S= distance of P from the directrix.
⇒ √(x−1)2+(y−1)2=x+y√2
⇒ 2x2+2y2−4x−4y+4=x2+y2+2xy
⇒ x2+y2−2xy−4x−4y+4=0
⇒ (x−y)2=4x+4y−4
⇒ (x−y)2=4(x+y−1)