The given equation of the parabola is in the standard form of a vertical parabola, which can be represented as y = ax². In this case, we can identify that a = 112.
To find the focus and directrix of a parabola given in this form, we can utilize the following key points:
- The vertex of the parabola is at the origin (0, 0) since there are no additional constants in the equation.
- The distance from the vertex to the focus, denoted as p, can be calculated using the formula p = 1/(4a).
Substituting our value of a:
p = 1/(4 × 112) = 1/448.
The focus of the parabola is located at (0, p). Thus, substituting our calculated value, the focus is at:
(0, 1/448).
Now, for the directrix, which is a line that is located p units away from the vertex in the opposite direction of the focus, we can determine its equation:
The directrix is given by the line y = -p. Therefore, substituting in our value of p gives:
y = -1/448.
In summary, for the parabola defined by the equation y = 112 x², the focus is at:
(0, 1/448)
and the directrix is:
y = -1/448.