|
|
|
|
![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
properties of parabola
|
(Topic)
|
|
|
Let us consider the parabola
 |
(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
 |
(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
 |
(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 $$y\!-\!y_0 \;=\; \frac{x_0}{p}(x\!-\!x_0) \;\equiv\; \frac{x_0x}{p}-2\!\cdot\!\frac{x_0^2}{2p} \;\equiv\; \frac{x_0x}{p}-2y_0.$$ This is simplified to
 |
(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 $$TF \;=\; y_0+\frac{p}{2} \;=\; P_0N \;=\; P_0F$$ (the last equality by the definition of parabola). Thus we see that the quadrilateral $P_0FTN$ 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.
- 1
- LAURI PIMIÄ: Analyyttinen geometria. Werner Söderström Osakeyhtiö, Porvoo and Helsinki (1958).
|
"properties of parabola" is owned by pahio.
|
|
(view preamble | get metadata)
Cross-references: focal radius, angles, apex, diagonals, rhombus, quadrilateral, equality, projection, ordinate, tangent of conic section, equation, slope, tangent, axis, parallel, line, chord, midpoint, arithmetic mean, sum, propertiess of quadratic equation, points, intersection, abscissas, quadratic equation, right hand side, parallel lines, cut, directrix, focus, distance, parametre, latus rectum, parabola
This is version 14 of properties of parabola, born on 2009-04-20, modified 2009-04-30.
Object id is 11749, canonical name is PropertiesOfParabola.
Accessed 684 times total.
Classification:
| AMS MSC: | 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry) | | | 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|
![\begin{pspicture}(-5,-3.5)(5,7) \psaxes[Dx=10,Dy=10]{->}(0,0)(-4.5,-3)(4.5,5) \r... ... \rput(-2.8,-1.3){directrix} \rput(-3,4){$y\,=\,\frac{x^2}{2p}$} \end{pspicture}](http://images.planetmath.org:8080/cache/objects/11749/js/img5.png "\begin{pspicture}(-5,-3.5)(5,7) \psaxes[Dx=10,Dy=10]{->}(0,0)(-4.5,-3)(4.5,5) \r... ... \rput(-2.8,-1.3){directrix} \rput(-3,4){$y\,=\,\frac{x^2}{2p}$} \end{pspicture}")