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

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”.
