Question

how to find an equation of parabola if vertex and latus rectum is given

Answer 1

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Step-by-step explanation:

Here's one approach.

Let the given information be a, b (the endpoints of the latus rectum) and v (the vertex).

The midpoint of the latus rectum is the focus.  So focus is at

f = ( a + b ) / 2

Let e be the point on the directrix collinear with the focus and the vertex.  Then the vertex is the midpoint between e and f.  So

e = 2v - f

A point x is on the parabola if its distance from f is equal to its distance from the directrix.  For the distance from the directrix, consider projecting x onto the axis.

Take n to be the unit vector in the direction from v to f.  So

n = ( f - v ) / | f - v |

Then the distance from x to the directrix is just the dot product

distance from x to directrix = ( x - e ) · n

The distance from x to f is also a (square root of a) dot product.

distance from x to focus = | x - f | = √( x - f ) · ( x - f )

The equation of the parabola is then given by

( x - f ) · ( x - f ) = [ ( x - e ) · n ]²

Of course, one would put in the values of e, f, n as computed above.

Notice this is general, so there is no restriction on the slope of the latus rectum.