What are the focus and directrix of the parabola with the equation y=1/12x^2
The equation of a parabola can be written in the form y = ax^2 + bx + c, where a, b, and c are constants. The equation you've provided, y = (1/12)x^2, is a simple form of a parabola with a = 1/12, b = 0, and c = 0.
For a parabola of the form y = ax^2 that opens upward or downward (as is the case with y = (1/12)x^2), the focus and directrix can be found using the formula 1/(4a). Since the vertex of this parabola is at the origin (0, 0), the focus and directrix are relatively simple to find.
The distance from the vertex to the focus (and to the directrix) in terms of the value 'a' is given by 1/(4a).
For the given parabola y = (1/12)x^2, we have a = 1/12. Therefore, the distance from the vertex to the focus and directrix is given by:
1/(4a) = 1/(4 * (1/12)) = 1/(1/3) = 3
Since the parabola opens upward, the focus will be at the point (0, 3) and the directrix will be the horizontal line y = -3, which is 3 units below the vertex.
In summary, the focus of the parabola y = (1/12)x^2 is at (0, 3), and the directrix is the line y = -3.