Given, focus of a parabola is at (0, 0) and vertex is at the intersection of line x + y = 1 and x - y = 3

Now solve these two equations, we get

x = 2, y = -1

So, vertex is at (2, -1)

Let (x₁ , y₁ ) be the coordinate of the point of intersection of the axis and the directrix. Then the vertex is

the mid-point of the line segment joining (x₁ , y₁ ) and the focus (0, 0)

So, (x₁ + 0)/2 = 2

=> x₁ = 4

and  (y₁ + 0)/2 = -1

=> y₁ = -2

Thus, the directrix meets the axis at (4, -2)

Let m is the slope of the axis, then

      m = slope of the line joining the focus and the vertex

=> m = (2 - 0)/(-1 - 0) = -2

So, the slope of the directix = -1/m = -1/(-2) = 1/2

Thus the directix pases through (4, -2) and has the slope 1/2

So, its equation is

      y + 2 = (1/2)(x - 4)

=> 2(y + 2) = x - 4

=> 2y + 4 = x - 4

=> x - 2y - 4 - 4 = 0

=> x - 2y - 8 = 0

Let P(x, y) be a point on the parabola. Then

Distance of P from the focus = Distance of P from the directrix

=> √{(x - 0)² + (y - 0)² } = |x - 2y - 8|/√{1² + (-2)² }

=> √{(x - 0)² + (y - 0)² } = |x - 2y - 8|/√(1 + 4)

=> √(x² + y² ) = |x - 2y - 8|/√5

Squaring on both side, we get

       x² + y² = (x - 2y - 8)² /5

=> 5(x² + y² ) = (x - 2y - 8)² 

=> 5x² + 5y² = x² + 4y² + 64 - 4xy + 24y - 16x

=> 5x² + 5y² - x² - 4y² - 64 + 4xy - 24y + 16x = 0

=> 4x² + y² - 64 + 4xy - 24y + 16x = 0

=> 4x² + y² + 4xy + 16x - 24y + 16x - 64 = 0

This is the required equation of the parabola.