Conics - what's a, what's b? (1 Viewer)

[astab]

Hey guys,

So I'm still unsure as to how you determine a and b in various conics equations/scenarios ie. b^2=a^2(1-e^2) for the ellipse and b^2=a^2(e^2-1) for the hyperbola. I understand that you let the "bigger value" represent a for the ellipse so that you get a fraction less than one when you solve for e, but what about the hyperbola - regardless of what a or b are, when you add one to the fraction b^2/a^2, the number will be bigger than one anyway? I tried a problem for a hyperbola with y intercepts (x^2/9+y^2/4=-1) and got a and b muddled up, which impacted the equation of my directices and foci. Sorry for the convoluted question. In short, how do you know what a and b are in different situations?

Thanks.

 

[Carrotsticks]

For the standard hyperbola(horizontal major axis), 'a' is below the x and 'b' is below the y.

For the standard ellipse (horizontal major axis), 'a' is below the x and 'b' is below the y.

Now for the conjugate hyperbola and the ellipse with vertical major axis, things get murky and different teachers use different methods.

Some leave 'a' and 'b' as always being below the x and y term respectively whereas others swap them.

 

[Ekman]

The way I look at it is:

In a hyperbola the 'a' is the one underneath the positive term and the 'b' is the one underneath the negative term. So in: The 'a' would be the root 6 and the 'b' would be 3.

In an ellipse, the 'a' is the largest value due to the formula: