a question in Conics (1 Viewer)

[Petyo]

It's question 20 page 89 in Fitzpatrick. I have attempted a few ways in approaching it but still I am stuck - -'.
If you have some time please take a look and help me. Any idea will be really appreciated.
this is the question:
P and Q are points on the ellipse (E) and the chord PQ contains a focus. Show that the line joining the point of intersection of the normals at P and Q and the midpoint of the chord PQ is parallel to the X-axis.
I have tried doing it this way: I tried to find the y-coordinate of the intersect of the 2 normals and then the y-coordinate of the midpoint of PQ and tried to prove these 2 are equal. It didn't work. = =!

 

[EvoRevolution]

P(x1,y1) and Q(x2,y2) are two points on the ellipse
Let the equation of the line PW be y = mx + k.
As PQ passes throuhg the focus (ae,0)

therefore mae + k therefore = -mae
y=mx-mae (equation of PQ)
Solve simultaneously the equations of chord PQ and the ellipse to find P and Q.

find midpoint of PQ

find the point of interesection of the normals of p and q

you have to use sum and product of roots in some parts of the solution

 

[Petyo]

wow thanks for your immediate response. btw what do you have for the point of intersection of the normals?

 

[EvoRevolution]

u find the equations of the normals at P and Q solve them simultaneously
then make y the subject.

equation normal: and same for the other one

when solvin sumlataneouly use the subtraction method easier

 

[gurmies]

In my opinion, you need a certain "condition" which you can acquire by using the fact that chord PQ contains a focus. I am currently trying this question, and from what I can see, it involves some heavy algebra xD. Hopefully i'll get back to you in a short while.

EDIT: This algebra is killing me -.-

 

[GUSSSSSSSSSSSSS]

well cos im a lazy bugger and cbb with all the algebra

just by lookin at the question, because it says parallel, you could possibly use similarity, i'd need to draw a diagram to be sure, however its definite you will have ONE common angle, however not sure how to incorporate any other information without drawing a diagram..

but then again it may be a complete waste of time lol

good luck..XDD

 

[MC Squidge]

ehhh ill drop a full solution tomorow

 

[Petyo]

THanks guys for all your replies.
@EvoRevolution: I've tried your way but it's getting quite complicated because the mid-point involves "m" (which I don't know how to get rid of afterward) while the intersection of the normals does not involve "m". maybe i'm wrong = =!
@gurmies: I'm not sure if I misunderstood the question but it seems to me that this question isn't just about algebra (I've been wrestling with it since this afternoon)
-----------------------------
Anyway, I've got y= [ aey1yo (x1-xo)] / [b^2 (yo - y1)]
and ae= (yoy1-y1xo)/(yo-y1) with (x1,y1) and (xo,yo) are the coordinates of Q and P...
but i reckon they're useless. :mad1:

 

[gurmies]

I just took a look at the published solutions. These guys have all the time in the world to publish the most elegant solution, and let me tell you - the solution is all algebra =( Download a copy and see: http://www.dingodemolitions.com.au/ext2/Fitzpatrick_solutions.zip

 

[Petyo]

eehh, THanks. Im done = =