Conic Sections
Parabolas
As we saw in Quadratic Functions , a parabola is the graph of a quadratic function. As part of our study of conics, we'll give it a new definition. A parabola is the set of all points equidistant from a line and a fixed point not on the line. The line is called the directrix, and the point is called the focus. The point on the parabola halfway between the focus and the directrix is the vertex. The line containing the focus and the vertex is the axis. A parabola is symmetric with respect to its axis. Below is a drawing of a parabola.
If a parabola has a vertical axis, the standard form of the equation of the parabola is this: (x - h)2 = 4p(y - k) , where p≠ 0 . The vertex of this parabola is at (h, k) . The focus is at (h, k + p) . The directrix is the line y = k - p . The axis is the line x = h . If p > 0 , the parabola opens upward, and if p < 0 , the parabola opens downward.
If a parabola has a horizontal axis, the standard form of the equation of the parabola is this: (y - k)2 = 4p(x - h) , where p≠ 0 . The vertex of this parabola is at (h, k) . The focus is at (h + p, k) . The directrix is the line x = h - p . The axis is the line y = k . If p > 0 , the parabola opens to the right, and if p < 0 , the parabola opens to the left. Note that this graph is not a function.
Let
P = (x, y)
be a point on a parabola. Let l be the tangent line to
the parabola at the point
P
. Let
be a line segment whose
endpoints are the focus of the parabola and
P
. Every parabola has the
following property: the angle
θ
between the tangent line l and
the segment
equal to the angle
μ
between the tangent line
and the axis of the parabola. This means (in a physical interpretation) that a
beam sent from the focus to any point on the parabola is reflected in a line
parallel to the axis. Furthermore, if a beam traveling in a line parallel to
the axis contacts the parabola, it will reflect to the focus. This is the
principle on which satellite dishes are built.
