Here's the story:
I am a high school student who absolutely loves math. So I took a university level mathematics course that is renowned throughout our school for being extremely rigorous and tough. Last week we had our first lesson, and this quarter we'll be focusing on analytical geometry. Our teacher, known at our school for being the best mathematician, gave us this as homework:
http://www.flickr.com/photos/78587548@N06/7944911996/in/photostream Find the cartesian equation of the ellipse
This probably is extremely easy for you guys, but don't forget this was our very first lesson. I couldn't solve it and today in class I found out nobody actually solved this. But what really shocked me, our teacher couldn't solve this either! He actually had no clue at all, he stared at it for like 20 minutes, and tried like FP=PQ but he couldn't solve it! So my question to you guys is, what is the solution (not only the answers but also explanation). I want to assure you guys that this is not our homework, and I want to know the answer as my 'smart' teacher doesn't know it either and I feel as if the only forum where people could answer these questions (with pleasure and) without a problem is this one.
PS – I am adding information as on the intial question there was a lot of uncertainty and I don't exactly know how this forum works (this is my first post), so here is some extra information:
- M (-a,0) and F (a,0)
- The ellipse is shown (approximately) by the ellipse in red, and, this is important: is a figure of all the points which are as far from the circle as from F (I'm Dutch, so the english terminalogy is tough, but I hope you know what I mean).
- r is the radius
- P (x,y)
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And there were 3 questions (but because he didn't even know the first one the 2 others were completely disregarded but these are the 3 questions):
- What is the cartesian equation (I believe it is called that) of the 'ellipse'
- What kind of figure would you get if you place F outside of the circle ( I believe he briefly said it would be a hyperbole but I'm not sure)
- And what would be the equation of that figure (the hyperbole).
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I'm extremely sorry for my English, I'm just so desperate to know the answer and not a single mathematics teacher at our school knows this.. it's a shame.
Best Answer
It's hard to tell from the picture exactly what you're using as the definition of the ellipse. One way of defining an ellipse is that the sum of distances from the foci to any point on the ellipse is constant. So let's say the foci are at $(a,0)$ and $(-a,0)$ and the sum of the distances is $d$. Then the point $(x,y)$ is on the ellipse if $$ \sqrt{(x-a)^2 + y^2} + \sqrt{(x+a)^2 + y^2} = d $$ Move the second square root to the right side, square both sides, and expand. You get $$ (x-a)^2 + y^2 = (x+a)^2 + y^2 - 2 d \sqrt{(x+a)^2 + y^2} + d^2 $$ After some more expansion, cancellation, and moving terms around, $$ 2 d \sqrt{(x+a)^2 + y^2} = d^2 + 4 a x $$ Again square both sides and expand: $$ 4\,{d}^{2}{x}^{2}+8\,{d}^{2}ax+4\,{a}^{2}{d}^{2}+4\,{d}^{2}{y}^{2}=16 \,{x}^{2}{a}^{2}+8\,{d}^{2}ax+{d}^{4}$$ Cancel the $8 d^2 a x$'s, bring the terms containing $x^2$ and $y^2$ to the left and the constant $4 a^2 d^2$ to the right, and collect terms: $$ (4 d^2 - 16 a^2) x^2 + 4 d^2 y^2 = d^4 - 4 a^2 d^2 $$