How did we arrive to the conculsion that the equation for an ellipse is $x^2/a + y^2/b = 1$?

Question

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?

Answer 1

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 $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Answer 2

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”.