High school mathematics question locus parabola?

Question

Hi guys im a bit stuck on my maths hw this is for locus parabola's

if a parabola has its vertex at the point (3,1) and focus at the point (3,3)

a) what is its focal length?

b) what is the equation of the directrix?

C) what is the equation of the parabola

can you also tell me how you got it in simple thx

Answer 1

a parabola is the locus of all points equidistant from a fixed point (focus) and a line (directrix).

I reccommend plotting the vertex and focus to get an idea what the parabola is doing.

a) the focal length is the distance from from the vertex to the focus. The focus is 2 units above the vertex. It is 2. And it's positive because we're going up (FROM the vertex TO the focus).

b) the directrix is the line, in this case, running horizontal underneath the parabola. By definition, the the vertex must be halfway in between the focus and directrix. If the focus was 2 above the vertex, the directrix must be 2 below the vertex. So the directrix is the line y = -1.

c) standard form...

(x - h)² = 4p(y - k)

(h, k) is the vertex and p is the distance from the vertex to the focus which we already found to be 2. So...

(x - 3)² = (4)(2)(y - 1)

(x - 3)² = 8(y - 1)

you can leave it like this. The standard form also varies a little depending on which way the parabola is opening.