first of all analyse the figure.

for the equation of a parabola, we need two important things.

1) the co-ordinates of focus

2) the equation of directrix.

Then, just equate the distance of any arbitratry point ( x, y ) on the parabola from directrix and the focus.

That is just the definition of the parabola.

Now the axis ofthe given parabola is perpendicular to the line y = x + 1

So the gradient of the axis = -1.

Again the axis passes through the origin ( 0, 0 ) which is also the focus.

So equation of axis => y = -x

Solving y = x + 1 and y = -x we get :-   x = -1/2 , y = 1/2.

So vertex of the parabola = ( -1/2, 1/2 )

So point of intersection of the axis and the directrix = ( -1, 1 ). This is because, vertex is the midpoint of the focus and the point where axis and directrix meet.

Slope of directrix = 1 ( since, parallel to y = x +1 )

Equation of directrix => y - 1 = 1. { x - (-1)} 

                                     => x - y + 2 = 0

So taking an arbitrary point (x, y ), we know that if its locus is to be that of a parabola, we must have :

Then square both sides to get the answer.

The answer will come out as :