What are the focus and directrix of the parabola with the equation y = x2?
The equation y = x² is a standard form equation of a parabola that opens upwards. The general form of a parabolic equation with vertex at the origin (0, 0) that opens upwards or downwards is given by:
y = ax² + bx + c, with a ≠ 0.
For the equation y = x²:
a = 1, b = 0, and c = 0.
Since b = 0 and c = 0, the vertex of this parabola is at the origin (0, 0). The focus of a parabola is a fixed point located inside the parabola, and the directrix is a line outside the parabola. Both are equidistant from any point on the parabola.
For a parabola with equation y = ax², where a > 0, the focus is located at the point (0, 1/(4a)), and the directrix is a horizontal line given by the equation y = -1/(4a).
Therefore, for y = x², we have a = 1, so:
The focus is at (0, 1/(4*1)), which simplifies to (0, 1/4).
The directrix is the line y = -1/(4*1), which simplifies to y = -1/4.
In summary:
Focus: (0, 1/4)
Directrix: y = -1/4