how to find the equation of directrix if vertex is(1,-1)&focus is(2,3)

vertex(1,-1) , focus(2,3)       (given)

let directrix cuts the axis of parabola at (h,k) then

vertex is midpoint of line joining focus & point at which directrix cuts the axis of parabola...

from above statement , (h+2)/2 = 1        &   (k+3)/2 = -1

                                      (h,k) = (0,-5)

slope of directrix is m & slope of axis is m¹ then

mm¹ = -1

m¹ = 4/1 = 4

m = -1/4

now eq of directrix : (y+5) = (x-0)(-1/4)

                           x+4y+20 = 0              .................1

since eccentricity of parabola is 1 so

  SP = ePM

distance of any point from directrix = e (distance of any point on parabola from focus)

x+4y+20/(17)^1/2 = [(x-2)²+(y-3)²]^1/2 

this is the required eq

approve if u like my ans

Approved