Why does the equation of an ellipse always have to equal 1?

Answer 1

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That is like asking why the equation of a line is always y = mx + c.

The answer is that a line doesn't have to use that form of an equation. x + 3y = 7 is a line. However, in math, we often try to create a standard equation for a specific type of problem and in the case of an ellipse it is equal to 1.

Why do we do that? We do it so that a) we can quickly recognize what type of shape this creates and b) we can use the same methods to solve/graph/manipulate these equations.

The short answer is that any ellipse can be represented as an equation = 1, but it is still an ellipse if an alternate version is used.

Answer 2

It doesn't
any equation with x^2 and y^2 on the same side, plus in between and different coefficients is an ellipse.

Having it = 1 means the numbers underneath the x^2 and y^2 are the squares of the intercepts so that makes sketching it easier.

Answer 3

It doesn't. It's just a convention.