2x^2 + 3x + 4y - 5 = 0
the vertex form of the equation of the parabola is equal to:
y = (-1/2) * (x + 3/4)^2 + 1.53125
the conic form of the equation of the parabola is equal to:
-2 * (y - 1.53125) = (x + 3/4)^2
the conics form is what you want.
the general form of the conics form is:
4p(y-k) = (x-h)^2
that's for a vertically aligned parabola which is what you have.
since the specific form of the conics form of this parabols is:
-2(y-1.53125) = (x+3/4)^2, then this means that:
4p = -2
h = -3/4
k = 1.53125
the vertex of the parabola is at (h,k) which is at (-3/4,1.53125).
the distance between the focus and the vertex of the parabola is equal to p.
since 4p = -2, this means that p = -2/4 which is equal to -1/2.
the distance between the vertex of the parabola and the line of the directrix is also equal to p.
the vertex of the parabola lies midway between the focus and the directrix.
the latus rectum in a parabola is equal to 4 * the length of the focus.
the length of the focus is equal to 2/4.
4 * that is equal to 8/4 which is equal to 2.
the latus rectum is the distance between the 2 points on the parabola that are on horizontal line that goes through the focus.
let's figure out some points.
the vertex is at (-3/4, 1.53125)
the focus is at 1.53125 - 1/2 which is at (-3/4, 1.03125).
the intersection of the line of symmetry of the parabola with the directrix is at 1.53125 + 1/2 which is at (-3/4, 2.03125).
the line of the directrix is therefore based on the equation of y = 2.03125.
the intersection of the latus rectum with the parabola will be at x = -3/4 plus or minus 1 and y = 1.03125. that makes the end point coordinates of the latus rectum at:
(-1.75,1.03125) and (.25,1.03125).
the following graph shows the relationship.
here are a couple of references you might find useful.
http://www.purplemath.com/modules/parabola.htm
http://www.mathwords.com/l/latus_rectum.htm
http://hotmath.com/hotmath_help/topics/latus-rectum.html