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    could anyone explain the highlighted part, why do things change if b > a? (why is the y-axis the major axis)
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    (Original post by thorn0123)


    could anyone explain the highlighted part, why do things change if b > a? (why is the y-axis the major axis)
    This is obvious.


    Set x and y, in turn, to zero and see how this condition affects the points at which the ellipse crosses the axes
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    (Original post by thorn0123)


    could anyone explain the highlighted part, why do things change if b > a? (why is the y-axis the major axis)
    Basically, the way you derive the equation for the ellipse starts with the assumption that the major axis is along the x-axis. The foci of an ellipse will always be along the major axis, so it is impossible to have foci at (ae,0) and (-ae,0) if the major axis is along the y-axis.

    Rather than having two equations to cover the two cases, you just note that a reflection in x = y changes direction of the major axis. If it was initially along the x-axis, reflection in that line will put it along the y-axis and vice versa. The reflection will necessarily change the position of the foci and directrices as well.

    If the equation was completely symmetric then it would not matter which of a and b was larger, but since they are different for an ellipse (excluding the case where it is actually a circle) it clearly matters which is larger. That lack of symmetry is really the key.

    Note that 0 < 1 - e^2 < 1 for 0 < e < 1 so you cannot have b^2 = a^2(1 - e^2) if b^2 > a^2. That would be a contradiction.
 
 
 
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