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#1 2015-01-10 19:20:49

Agnishom
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From: Riemann Sphere
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Focal Chords of an Ellipse

PSQ is a focal chord of the ellipse 4x^2 + 9y^2 = 36 such that SP = 4.

If S' is another focus, write the value of S'Q.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
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#2 2015-01-10 22:16:40

Bob
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Re: Focal Chords of an Ellipse

hi Agnishom,

Here's a few properties of the ellipse that should help:

(1) The major axis goes from (-3,0) to (+3,0)

(2) The minor axis goes from (0,-2) to (0,+2)

(3) The foci are at (-f,0) and (+f,0) where f^2 = 3^2 - 2^2

(4) S'R + RS = constant for all R on the ellipse.  As (3,0) lies on the ellipse you can work out the constant, and hence S'P.

(5) You can use this to show S'PS = 90.

(6) SQ + QS' = same constant.

(7)  The parametric equations are x = 3cos(θ) and y = 2sin(θ)

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#3 2015-01-10 23:57:40

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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Re: Focal Chords of an Ellipse

How do I locate the other end of the chord?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#4 2015-01-11 01:09:06

Bob
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Re: Focal Chords of an Ellipse

Hi,

I wasn't fully sure when I posted earlier, but, as often happens to me, my ideas have become clearer since.

You know SP and you can calculate S'P from the constant.

You also know that SQ + QS' comes to the same constant.

With these values it turns out that S'PS is a right angle and so the triangle S'PQ is right angled.  So you can form a second equation involving S'Q and SQ.

Eliminate SQ and you can find S'Q.

On Sketchpad I get about 5.17, but I'm not very confident of the accuracy as I cannot force Q to be on PS and on the curve.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#5 2015-01-11 02:12:24

Agnishom
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From: Riemann Sphere
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Re: Focal Chords of an Ellipse

I'm getting 5.7 too. Thanks.

What if SPS' were not right?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#6 2015-01-11 02:17:36

Bob
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Posts: 10,053

Re: Focal Chords of an Ellipse

cosine rule.

B


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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#7 2015-01-11 02:42:42

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,974
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Re: Focal Chords of an Ellipse

Cosine Rule to compute what?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#8 2015-01-11 04:27:54

Bob
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Registered: 2010-06-20
Posts: 10,053

Re: Focal Chords of an Ellipse

Well it's the same thing really. 

S'P + PS is known.  SP is known.  S'S is known.  So you can calculate angle S'PS.

Now work on triangle S'PQ.

You know S'P, PQ = PS + SQ where SQ = constant - S'P

The angle S'PQ is the same as S'PS, so you can get an equation for S'Q with only SQ unknown.  Using these two you can get S'Q as before.  It's just a trickier calculation.

Bob

ps.  If you use the parametric form the equation for a chord is simpler than the rectangular form so this might give an alternative method.  I started this, but got tied up in trig. and algebra which is  why I switched to using the points to foci = constant rule as it looked (and indeed was) simpler.


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob smile

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