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# parabola

A *parabola* is the locus of points $P$ in the Euclidean plane
which are equidistant from a given line $\ell$, called the
*directrix*, and a given point $F$ not on the directrix, called
the *focus*.

To obtain a simple equation for the parabola, assume that the directrix is parallel to the $x$-axis, the focus is on the $y$-axis, and the directrix and focus are the same distance from the origin. By reflecting the plane if necessary, this means that there is a positive number $a$ such that the equation of the directrix $\ell$ is $y=-a$ and the position of the focus $F$ is $(0,a)$. Then the condition that a point $(x,y)$ is equidistant from $\ell$ and $F$ can be interpreted as the equation

$(y+a)^{2}=x^{2}+(y-a)^{2}.$ |

Since $(y+a)^{2}-(y-a)^{2}=4ay$, the above equation simplifies to

$y=\frac{1}{4a}x^{2}.$ |

Below is the graph of a parabola for $a=1$:

From the equation

$y=\frac{1}{4a}x^{2},$ |

we can immediately observe some important
properties of the parabola. First, since $x^{2}$ is an even function,
the parabola is symmetric with respect to the $y$-axis; this can also
be deduced directly from the geometric definition of the parabola. The intersection point of the parabola and the symmetry axis, is called the *apex* of the parabola; it is the point of the parabola nearest the directrix.
Second, notice that the coefficient of $x^{2}$ in the equation of the
parabola is inversely proportional to $2a$, the distance between the
focus and the directrix. So this distance controls how rapidly the
function $\frac{1}{4a}x^{2}$ grows. As $a$ tends to zero, the parabola
becomes flatter and flatter, tending to the straight line $y=0$ in
the degenerate case $a=0$. On the other hand, as $a$ increases, the
curvature of the parabola at $0$ increases. When $a$ tends to
infinity, the parabola begins to resemble a hairpin more and more
until it suddenly becomes a single point, the origin, in the degenerate
case $a=\infty$.

The parabola is a conic section with eccentricity 1. All parabolas are similar, which follows directly from the definition of parabola.

## Mathematics Subject Classification

53A04*no label found*51N20

*no label found*

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