Which is the equation of a parabola with a directrix at y = −3 and a focus at (5, 3)?
y = 1/12 (x-5)^2
y = -1/12 (x-5)^2
y = 1/12 (x+5)^2
y = -1/12 (x--+5)^2

y = 1/12 (x − 5)²

Step-by-step explanation:

We can solve this graphically without doing calculations.

The y component of the focus is y = 3.  Since this is above the directrix, we know this is an upward facing parabola, so it must have a positive coefficient.  That narrows the possible answers to A and C.

The x component of the focus is x = 5.  Since this is above the vertex, we know the x component of the vertex is also x = 5.

So the answer is A. y = 1/12 (x−5)².

But let's say this wasn't a multiple choice question and we needed to do calculations.  The equation of a parabola is:

y = 1/(4p) (x − h)² + k

where (h, k) is the vertex and p is the distance from the vertex to the focus.

The vertex is halfway between the focus and the directrix.  So p is half the difference of the y components:

p = (3 − (-3)) / 2

p = 3

k, the y component of the vertex, is the average:

k = (3 + (-3)) / 2

k = 0

And h, the x component of the vertex, is the same as the focus:

h = 5

So:

y = 1/(4×3) (x − 5)² + 0

y = 1/12 (x − 5)²

1/800 is the right answer