The positive number a is called the focal length of the parabola. If you have the equation of a parabola in vertex form y a x h 2 k then the vertex is at h k and the focus is h k 1 4 a. The formula for the arc length of a parabola is.
The Arc Length of a Parabola Let us calculate the length of the parabolic arc y x2.
Axis of symmetry. According to the arc length formula La Z a 0 p 1 y0x2 dx Z a 0 p 1 2x2 dx. L is the length of the parabola arc. The same goes for all of the other distances from a point on the parabola to the focus and directrix l_2 l_3 text etc.