Question

Derive the equation of the parabola with a focus at (6, 2) and a directrix of y = 1.

Answer 1

First find the k value for the vertex. So take half the distance between the directrix and the focus.

So (2+1)/2 = 1.5 The vertex is (6,1.5)

We know that the parabola opens up, since the directrix , y=1 is less than 2 value of the focus where the directrix is parallel to the x axis.

So the equation for a parabola opening upwards is\[(x-h)^{2}= 4p(y-k)\]

Where p is the distance between the vertex and the focus, or distance between the vertex and the directrix.

Solve for p. So 2-1.5=.5

Substituting in the equation we get.
\[(x-6)^{2} = (4)(.5)(y-1.5)\]
Expanding the equation and combining terms, we get.\[x ^{2}-12x+36=2y-3\]\[x ^{2}-12x-2y+39=0\]