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# properties of parabola

$\displaystyle y\;=\;\frac{x^{2}}{2p}$ | (1) |

where $2p$ is the latus rectum (a.k.a. parametre), i.e. the double distance of the focus from the directrix.

1. Cut the parabola with the family

$\displaystyle y\;=\;mx\!+\!k$ | (2) |

of parallel lines ($m$ is constant). Substituting the right hand side of (2) into (1) yields the quadratic equation

$x^{2}-2mpx-2kp\;=\;0$ |

which determines the abscissas of the intersection points. By the properties of quadratic equations, the sum of abscissas of both points is $2mp$ and thus their arithmetic mean is $mp$. This means that the midpoint of the chord cut by the parabola from the line (2) has the constant abscissa

$\displaystyle x_{0}\;=\;mp.$ | (3) |

Accordingly, *all midpoints of parallel chords of parabola are on a line parallel to the axis of parabola*.

2. In the case the line (2) is a tangent of the parabola, the midpoint $P_{0}$ of the chord coincides with the point of tangency, having the abscissa $x_{0}$. Thus the slope of the tangent is by (3) equal

$m_{t}\;=\;\frac{x_{0}}{p}$ |

and therefore the equation of the tangent is

$y\!-\!y_{0}\;=\;\frac{x_{0}}{p}(x\!-\!x_{0})\;\equiv\;\frac{x_{0}x}{p}-2\!% \cdot\!\frac{x_{0}^{2}}{2p}\;\equiv\;\frac{x_{0}x}{p}-2y_{0}.$ |

This is simplified to

$\displaystyle y\!+\!y_{0}\;=\;\frac{x_{0}x}{p}$ | (4) |

(cf. tangent of conic section).

3. The tangent (4) cuts the axis of the parabola in the point $T$ whose ordinate is $-y_{0}\,(\leqq\,0)$. If $F$ is the focus and $N$ the projection of $P_{0}$ on the directrix, we have

$TF\;=\;y_{0}+\frac{p}{2}\;=\;P_{0}N\;=\;P_{0}F$ |

(the last equality by the definition of parabola). Thus we see that the quadrilateral $P_{0}FTN$ is a rhombus. Therefore its diagonals bisect and intersect each other perpendicularly (in the point $M$ on the $x$-axis). Consequently, we get the following two results.

*The projection of the focus on any tangent of parabola is on the tangent whose point of tangency is the apex of the parabola.*

*The tangent of parabola forms equal angles with the axis and the focal radius drawn to the point of tangency.*

# References

- 1 Lauri Pimiä: Analyyttinen geometria. Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).

## Mathematics Subject Classification

53A04*no label found*51N20

*no label found*

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