how to prove reflection property!!! (1 Viewer)

[heath]

our teacher showed us this reallly really long method of proving the reflection property of the ellipse and hyperbola.

i was wondering if there was an easy, preferably, quick method.


???

 

[Affinity]

method 1 gradient bash:

find gradients of teh two lines from focii to the point
find gradient of ellipse at the point
use the tan(@) =| (m1 - m2)/1 + m1m2 | formula

method 2 'smart' way:

prove SP + SP' = 2a for any point P on teh ellipse

show that the sum of distances SQ + QS' where Q is any point on the tangent l at P is minimized when Q is P.

let R be the reflection of S about l

the PR = PS and show that S'PR is a straight line

use geometry to deduce the angles are equal

 

[felix_js]

i remember seeing a way to do it with circles, can't remember it off the top of my head but im sure it was from the arnold book