Find the directrix of the parabola 12(y+4)=(y-6)^2?

Answer 1

OK, I'm sure you mean either:

12(y + 4) = (x - 6)²

Or

12(x + 4) = (y - 6)²


I'll do both.

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12(y + 4) = (x - 6)²:

y + 4 = 1/12(x - 6)²

y = 1/12(x - 6)² - 4

This a parabola opening upwards, with vertex at (6, -4).

The number outside of the parentheses (1/12) is the important number for finding the focus and directrix. The key number fo find is "p", using the following formula:

1/(4p) = 1/12
p = 3

So, knowing p, and knowing that the parabola opens upwards, we now know that the focus is 3 units above the vertex F = (6, -1), and the directrix is the horizontal line 3 units below: y = -7.

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12(x + 4) = (y - 6)²

x + 4 = 1/12(y - 6)²

x = 1/12(y - 6)² - 4

Everything from above is the same, except that now you add p to the x-coordinate of the vertex, and the directrix is vertical. So the vertex is (-4, 6); the focus is (-1, 6), and the directrix is x = -7.