1. ## Ellispe Eccentricity error

we know
b² = a²(1-e²)

and the equation for an ellipse is

x²/a² + y²/b² = 1

We're told the Ellipse E has equation

x²/2² + y²/3² = 1

so 9 = 4(1-e²)
4e² = -5

which you cant do, because that would be a complex number. BUT howcome you can get around this by swapping the values for a and b round. That also means that the focus has now transferred to the y axis, rather than the x, as foci as (ae,0) and now a = b?

Thanks

2. An ellipse has always a major axis and a minor axis. And the foci lie on major axis.

In your question, the major axis is y-axis and its length is 6. The minor axis is x-axis and its length is 4.

Formula for eccentricity:
$\displaystyle 1-e^2=\frac{b^2}{a^2}$
where b is length of minor axis
and, a is length of major axis

And the focus:$\displaystyle ae$ where a is the length of major axis.

The formula usually given in textbooks assume that a is the major axis whereas in your question a is the minor axis...hence we have to swap a and b