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