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How to find the length of latus rectum of a parabola?

The length of the latus rectum is equal to the distance between the focus and vertex of the parabola. It is also equal to the perpendicular distance of the vertex from the directrix of the parabola.

Hence if you know the coordinates of the vertex and focus of a parabola, you can find the length of its latus rectum by using the distance formula as explained in the following example:

Example: A parabola has vertex (0, 3) and focus (0, 6).

By using distance formula, distance between the focus and vertex is:
`= sqrt((6 - 3)^2 + (0 - 0)^2)`
`= 3` 
Hence length of the latus rectum is equal to 3.

Simple. Isn't it?

But suppose you are given the equation of the parabola, then the method for finding the latus rectum becomes a bit different. Of course, you can find the vertex and focus coordinates from the equation and then use the distance formula, as above, to find the length of the latus rectum, but the following procedure describes how you would just find the length of the latus rectum from the equation of the parabola:

The length of the latus rectum is equal to four times the distance between the focus and the vertex. Thus in order to find the length of the latus rectum, first you have to determine the distance between the focus and the vertex of the parabola.

Thus if you know the coordinates of the vertex and focus of a parabola, you can simply use the distance formula to find the distance between the two points, then multiply it by four to get the length of the latus rectum.

On the other hand, if the equation of the parabola is given, you have to follow the following steps to find the length of the latus rectum:

Example: Suppose we have the equation `x^2 = 4ay`. 

It's graph is as follows:


In this equation 'a' is a constant value and is equal to the distance between the focus and vertex of the parabola.