Let A(2, -3) be the vertex and F(0, 5) be the focus.
Let the axis meet the directrix at a point Z(α,β).
Then, A is the mid point of ZF.
So, 2 = α+0/2
Thus, the coordinates of Z are (4, -11).
Now, slope of ZF = 5+11/0-4=-4
So, slope of the directrix = 1/4
Thus, the directrix is a line passing through the point (4, -11) and has a slope equal to 1/4.
Thus, the equation of the directrix is given by:
Let P (x, y) be any point on the parabola. Then,
PF = length of the perpendicular from P on x-4y-48=0
Simplify this expression to get the required equation of the parabola.
We hope that this clarifies your query.