Definition 10.1.4. Parabola.
A parabola is the locus of all points equidistant from a point (called a focus) and a line (called the directrix) that does not contain the focus.
Now square both sides.
\begin{align*} x^2+(y-p)^2 \amp = (y+p)^2\\ x^2+y^2-2yp+p^2 \amp = y^2+2yp+p^2\\ x^2 \amp =4yp\\ y\amp = \frac{1}{4p}x^2\text{.} \end{align*}Any ray emanating from the focus that intersects the parabola reflects off along a line perpendicular to the directrix.
Horizontal Transverse Axis |
Vertical Transverse Axis | |
\(\ds y=\pm\frac ba(x-h)+k\) | \(\ds y=\pm\frac ab(x-h)+k\text{.}\) |
Planet | Distance from center to vertex |
Orbit eccentricity |
Mercury | \(0.387\) A.U. | \(0.2056\) |
Earth | 1 A.U. | \(0.0167\) |
Mars | \(1.524\) A.U. | \(0.0934\) |