why is it that the latus rectum is equal to 4a?

Answer 1

This question has been up here for a while without, in my opinion, a satisfactory answer, so I'll take a stab at it.

There are both geometric and algebraic explanations, and the geometric one is easier. Begin with one definition of a parabola: the locus of points equidistant from a fixed point (the focus) and a fixed line (the directrix).

Consider a parabola opening upward with vertex at (h,k) and focal length a. Then the focus is at (h,k+a), and the directrix is the line y = k-a. The so-called "line of symmetry" passes through both the focus and vertex, and is perpendicular to the directrix. The vertex is the midpoint of the segment connecting the directrix and the focus, and the length of that segment is 2a.

The latus rectum passes through the focus and is parallel to the directrix. The equation of that line is y = k, and the distance between the latus rectum and directrix is 2a.

The ends of the latus rectum lie on the parabola, and from the definition of the parabola, they must each lie a distance 2a from the focus. Therefore, the ends of the latus rectum are at (h-2a, k) and (h+2a, k).

The distance between those two endpoints is (h+2a) - (h-2a) = 4a, which is four times the focal length.

That's the geometric explanation.
...............

Here's an algebraic example taken from the problems section of a textbook. (Beware -- it has a lot of fractions.)

Find the vertex, focus, directrix, and latus rectum of

y = 4x^2 + 6x + 2

We need to get it in this form:

(x - h)^2 = 4a(y - k)

Begin by transposing some terms and preparing to complete the square in x:

4x^2 + 6x = y - 2
4(x^2 + 3/2 x + ...) = y - 2 + ...
4(x^2 + 3/2 x + 9/16) = y - 2 + 9/4
4(x + 3/4)^2 = y + 1/4
(x + 3/4)^2 = 1/4 (y + 1/4) (Standard form)

The vertex is at (-3/4, -1/4).
The focal length a = 1/16, so the focus is at (-3/4, -3/16). (Note that -1/4 + 1/16 = -3/16.)

The latus rectum lies along the line y = -3/16, so we can plug that value into the standard form equation and solve for x to get the ends of the latus rectum. For y = -3/16:

(x + 3/4)^2 = 1/4 (-3/16 + 1/4) = (1/4)(1/16) = 1/64
x + 3/4 = +/- 1/8
x = -3/4 +/- 1/8 = (-6 +/- 1) / 8 = -7/8 or -5/8

Those are the x-values at the ends of the latus rectum. The length of the latus rectum is -5/8 - (-7/8) = 1/4, which is four times the focal length (a = 1/16, above).

Now you have both a geometric and algebraic explanation. There's also another algebraic explanation that's more difficult, but I won't do that one.

Answer 2

You probably know what latus rectum is, so maussy's answer is most probably not enlightening.

Do you want a proof? If so, there are numerous approaches to it; depending on where you are allowed to start. Is this for a class? What do you officially "know" or are allowed to "use" to prove it? My point is, if you start with the "right" definition of a parabola, the proof is OK. If you start with the wrong one [which happens to be the one normally taught in most schools, I believe], the full proof is lengthy.

So I am not going to write the proof down; we need some input from you first...

Answer 3

answer in my link