Find the standard equation of the parabola with the focus at

Find the standard equation of the parabola with the focus at (0, 3) and the focus at (0, 3) and the directrix, the line y = -3. Give the two points that define the latus rectum.

Solution

focus=(0,3) and directrix ,y=-3

The standard equation of parabola is

(x-h)²=4p(y-k)

focus =(0,3)

that is k+p=3

and directrix ,y=-3

that is k-p=-3

So we have two equations

k+p=3 and k-p=-3

Solving them we get k=0 and p=3

Therefore required equation is

(x-0)²=4(3)(y-k)

x^2=12y

End points of latus rectum are (h+p,k+-2p)=(0+3,0+-2(3))=(3,+-6)

therefore end points are (3,6) and (3,-6)