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so I have this problem for my homework:

Consider the elipse: $\dfrac {x^2}{a^2} + \dfrac{y^2}{b^2}=1$ where $0<b<a$. For every point (x,y) on the ellipse find the the perpendicular line to the ellipse so that the point $(x,y)$ is on that line. This line cuts the ellipse in another point $(x',y')$. Prove that the distance between these two points is $$D(x,y)=2\dfrac {(\dfrac {x^2}{a^4} + \dfrac{y^2}{b^4})^{3/2}}{\dfrac {x^2}{a^6} + \dfrac{y^2}{b^6}}\ .$$ So I've alrredy done that; but after that it says:

Use Lagrange Multipliers to minimize the function $D(x,y)$. But the equations get complicated and messy and I dont know if i have to consider the line or just the ellipse. Logically the minimum is at $(0,b)$.

If anyone can help me I would thak you so much!

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but 0 isn't in the elipse, the profersor wants ou to find the pont (x,y) on the elipse such that the distance to (x',y') is minumun –  user1080987 Dec 3 '12 at 14:30
    
I have edited your question slightly. In particular it reads now $(x,y)$ is on the ellipse and not "in" the ellipse. –  Christian Blatter Dec 3 '12 at 14:45
    
Doing a quick experiment using Cinderella, it seems that your intuition about what minimum point you expect is wrong: there are points off the axes of symmetry where the distance becomes a lot smaller. The larger the excentircity, the more pronounced this effect becomes. –  MvG Dec 3 '12 at 17:30
    
en.wikipedia.org/wiki/Lagrange_multipliers describes how to maximize $f$ subject to $g=c$. So you'd have $g=c$ as the formula of your ellipsis, and $-D$ as the thing you want to maximize. Now plug these things into the formulas on Wikipedia, compute three derivatives, and you should have a set of three equations. Are you allowed to do this using a computer algebra system? If not, do things become messy enough to actually prevent a manual solution? If so, then re-examine your term above (which I have not checked) and see whether there might be some error in there. –  MvG Dec 3 '12 at 17:39
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1 Answer

I think it is not necessary to use Lagrange Multipliers. we can replace $\dfrac{x^2}{a^2}$ with $1- \dfrac{y^2}{b^2}$, then we have:

$D(y)=2\dfrac{(\dfrac{1}{a^2}(1- \dfrac{y^2}{b^2})+ \dfrac{y^2}{b^4})^\frac{3}{2}}{\dfrac{1}{a^4}(1- \dfrac{y^2}{b^2})+ \dfrac{y^2}{b^4}}=2\dfrac{(C_1+C_2y^2)^\frac{3}{2}}{C_3+C_4y^2}$, and $C_1=\dfrac{1}{a^2},C_2=\dfrac{1}{b^2}(\dfrac{1}{b^2}-\dfrac{1}{a^2})=\dfrac{a^2-b^2}{a^2b^4}$,$C_3=\dfrac{1}{a^4},C_4=\dfrac{1}{b^2}(\dfrac{1}{b^4}-\dfrac{1}{a^4})=\dfrac{a^4-b^4}{a^4b^6}$, let $z=y^2$,we have a very simple fomula:

$D(z)=2\dfrac{(C_1+C_2z)^\frac{3}{2}}{C_3+C_4z}$

$D'(z)=2*(\dfrac{3}{2}C_2\dfrac{(C_1+C_2z)^\frac{1}{2}}{C_3+C_4z}-C_4\dfrac{(C_1+C_2z)^\frac{3}{2}}{(C_3+C_4z)^2})$

$=2*\dfrac{(C_1+C_2z)^\frac{1}{2}}{C_3+C_4z}*(\dfrac{3C_2}{2}-C_4*\dfrac{C_1+C_2z}{C_3+C_4z})$,

since $\dfrac{(C_1+C_2z)^\frac{1}{2}}{C_3+C_4z}>0$, when $D'(z)=0$, we have:

$\dfrac{3C_2}{2}-C_4*\dfrac{C_1+C_2z}{C_3+C_4z}=0$, then we put all staff in, we get: $D'(z)=F(z)*Q(z), F(z)>0$,$Q(z)=b^6-2a^2b^4+(a^4-b^4)z$ , if $D'(z)=0$,then $Q(z)=0$,

$z=\dfrac{b^4(2a^2-b^2)}{a^4-b^4}$, now here is a trick:

$z=y^2 \leq b^2$ then $\dfrac{b^4(2a^2-b^2)}{a^4-b^4} \leq b^2$,ie.$a^2 \geq 2b^2$ it means only under this condition, $D'(z)=0$ can be satisfied.

when $a^2 \geq 2b^2$, we have:

$Q(z)=b^4(b^2-z)+a^2(a^2z-2b^4), $when $z=b^2, Q(z) \geq 0$ ie $D'(z)\geq 0$;,when $ z=0, Q(z)=b^4(b^2-2a^2) < 0$,ie $D'(z)<0$ , which means D(z) has min. put $z=\dfrac{b^4(2a^2-b^2)}{a^4-b^4}$,we get:

$ D_{min}(z)=\dfrac{\sqrt{27}a^2b^2}{(a^2+b^2)^\frac{3}{2}} \leq 2b $

when $a^2 < 2b^2$, we have:

$Q(z)=a^2z(a^2-2b^2)+b^2(2a^2-b^2)(z-b^2)<0$, which mean when $z=b^2$, $D(z)$ has its min which is $2b$.

BTW, this is a Japanese Temple Geometry problem which discussed about 300 years ago.

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It looks easier to minimize $D^2(x, y)$, since the square roots go away and the resulting $x$ and $y$ will be the same. –  marty cohen Jun 16 '13 at 6:43
    
You might well think so. I tried that out thinking there'd be a cleaner and quicker way to deal with the algebra. Alas, it is tidier but doesn't reduce the amount of expression-wrangling... –  RecklessReckoner Feb 20 at 8:24
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