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Usually in textbooks, it is just given and said to memorize it as it is. What is the reason for an ellipse to have an unique graph equation, that involves this specific terms? How did the formula come into existance in the first place? Can I get the idea on how the derivation was done?

Rócherz
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Joe Guanyu
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    Sure. It starts with what you want the word "ellipse" to mean, and works its way to that equation. The exact derivation depends entirely on what you want "ellipse" to mean in the first place. What does "ellipse" mean to you? Also, do you accept that $x^2+y^2=1$ yields a circle, or would we need to derive that too should we need it? – Arthur Nov 01 '22 at 05:36
  • A curve that has two axises. it is the superset of circles. – Joe Guanyu Nov 01 '22 at 05:43
  • Can you please show me the derivation of circles. Also please excuse my ignorance in general topics of math, I am just highschooler trying to learn something new. – Joe Guanyu Nov 01 '22 at 05:46
  • Just do a web search for “ellipse derivation” or something. See here, for example: https://mathworld.wolfram.com/Ellipse.html – Hans Lundmark Nov 01 '22 at 05:48
  • Thanks a lot. It is a bit difficult to understand but I will try my best. – Joe Guanyu Nov 01 '22 at 05:51
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    A standard ellipse is obtained from a unit circle (which has equation say $x^2+y^2=1$) by stretching along one axis by a factor of $a$ and along another axis by a factor of $b$. That immedietaly gives your equation $(x/a)^2+(y/b)^2=1$. Of course now you have to prove that this construction agrees with other geometric or analytic definitions of an ellipse, but that is another story (and you don't mention any favourite definition you would like to compare with), so I stop here.. – Michal Adamaszek Nov 01 '22 at 07:32

2 Answers2

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We start from finding the equation of a circle. A circle is the collection of points of radius $R$ from its center.

We therefore start with a circle of radius $R$ from the origin $r^2=R^2$

Then, we use the Pythagorean Theorem and find that $r^2=x'^2+y'^2$, which gives us $x'^2+y'^2=R^2$.

Now that we have the equation for a circle, we now move onto an ellipse. An ellipse is a circle that has been stretched nonuniformly. Without loss of generality, we will start with the unit circle and stretch it by a factor of $a$ in the $x$ direction and $b$ in the $y$ direction, Then, substitute $x'=\frac xa$ and $y'=\frac yb$, and get the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Moko19
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You have to define what you mean by “ellipse”, and then you can derive an equation from your definition.

One possible definition is: given two points A and B, an ellipse is the locus of points P such that the distance from P to A plus the distance from P to B is a constant.

You can draw the ellipse by putting a loop of string around the points A and B.

Using this definition, you can derive the usual ellipse equation.

More details here or here. Or just search for “ellipse pins string”.

bubba
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