conics query. (1 Viewer)

[cwag]

I've never fully understood...that when calculating eccentricity for either ellipse or hyperbola, there are certain situations when the roles of a and b swap around....What are these situations?....for example, wheb b>a for ellipse i think the eccentricty equation becomes a²=b²(1 - e²)...and focus is be, directrix b/e...however there are other times when is just the normal b²=a²(1 - e²)..plz explain?

 

[tommykins]

回复: conics query.

If b>a, the ellipse is an egg with the major axis on the y axis and the minor axis on the x axis, hence why the b and a value changes in the formula.

If a>b, then it's your typical watermlon (Looool) and the "typical" equation holds

 

[cwag]

yea but what about for hyperbola? does it change for y²/a² - x²/b² =1....or when b>a?

 

[cwag]

does anybody know this?? i was thinking about it, and maybe its only for ellipse, because when determining eccentricity e²= 1 - b²/a²

and since e is real, b²/a² must be < 1 so if b>a then the roles of a and b switch..

however, for a hyperbola e² = 1 + b²/a², which really doesn't matter how big or small because its always > 0 so e is real.

If any of that made sense, im still unsure as to whether the roles of a and b switch for a different situation...like y²/b² - x²/a² = 1?

 

[nrs1990]

For the hyperbola, they can switch, but it has nothing to do with if b>a. You switch a and b around when you're working with the inverse function, that is, when y<SUP>2</SUP>/b<SUP>2</SUP> - x<SUP>2</SUP>/a<SUP>2</SUP> = 1 . That hyperbola goes up and down, rather than left and right.