why the length  of latus rectum is 4a ?

 Hi!

Here is the answer to your query.

A double ordinate that passes the focus is called the latus rectum i.e. the latus rectum of a parabola is a chord passing through the focus perpendicular to the axis.

In the given figure LSL' is the Latus rectum of the parabola y² = 4ax. 

By the symmetry of the curve SL = SL' = λ (Say). So the coordinates of L are (a, λ).

Since L lies on y² = 4ax, therefore

λ² = 4a² ⇒ λ = 2a ⇒ LL' = 2λ = 4a

∴ Latus rectum = 4a

and the coordinates of L and L',  points of latus rectum are (a, 2a) and (a, – 2a) respectively.

Cheers!