The standard equation of a parabola with focus (2,-3) and directrix x = 6 is?

Question

The answer says it's (x - 4) = -8(y + 3)^2

but for x - orientation, the equation is (y - k)^2 = 4p (x - h). Why is 4p distributed to (y - k)?
not to (x - h)?

Answer 1

The standard form of horizontal parabola (y-k)² = 4p(x-h) (where p not equals to 0.)

Vertex of parabola (h,k) and directrix line is x = h - p.

directrix x = 6 , focus = (2, -3)

Focus = (2, -3)

The focus and the directrix were given, but not the vertex.

The vertex is halfway between the focus and the directrix.

In this case, we have to use the midpoint formula.

M = [(x1+x2)/2),((y1+y2)/2]

M =  [(2+6)/2 , (-3-3)/2]

(h , k) = (4, -3)

vertex (4, -3)

We need to find p. Use the formula for calculating the directrix.

Directrix x = h - p

6 = 4 - p

-p = 6 - 4

p = -2

p < 0 so parabola opens left.

Substitute the values p , (h, k) in (y - k)² = 4p(x - h).

(y - (-3))²  = 4(-2)(x - 4)

The answer is -8(x - 4) = (y + 3)².

In this case given the directrix x = 6

The directrix is x = h - p ,so here 4p is distributed for (y - k).

Here the equation is horizontal parabola standard form (y - k)² = 4p(x - h).

If given the directrix is y = 6

then directrix y = k - p , so here 4p is distiributed for (x - h)

Here the equation is vertical parabola standard form(x - h)² = 4p(y - k).