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Mathematik-Online problems:

Problem 87: Focus and Reflexion of Light in Ellipses

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
Given the ellipses $ {\cal{E}}:$ $ {\displaystyle{\frac{x_1^2}{a^2}+\frac{x_2^2}{b^2}}}$ $ \,-1=0$ , with $ a>b$ in $ \mathbb{R}^2$ .
The points $ F_1=(-c, 0)$ and $ F_2=(c, 0)$ , with $ c=\sqrt{a^2-b^2}$ are called foci of $ {\cal{E}}$ . The value $ \varepsilon=c/a$ is called eccentricity and the lines $ \ell_{1/2}: \ x_1=\pm\,a^2/c$ are called guidelines of $ {\cal{E}}$ . Proof:
a)
$ d\,(P,F_1)+d\,(P,F_2)\,=\,2a$ , for each point $ P\in{\cal{E}}$ .
b)
$ {\displaystyle{\frac{d\,(P,F_1)}{d\,(P,\ell_1)}}}$ $ =$ $ {\displaystyle{\frac{d\,(P,F_2)}{d\,(P,\ell_2)}}}$ $ = \, \varepsilon$ , for each point $ P\in{\cal{E}}$ .
c)
Each ray which is starting in $ F_1$ and reflected about $ {\cal{E}}$ arrives the point $ F_2$ , and contrary.

(Authors: Kimmerle/Apprich/Höfert)

Solution:

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  automatisch erstellt am 14. 12. 2007