Difference between ellipse circle parabola and hyperbola equations

The equation of a circle is:

(x - h)² + (y - k)² = r²

The equation of a parabola is:

(x - h)² = 4p(y - k)

The equation of an ellipse is: 

[tex] \frac{(x-h) ^{2} }{ a^{2} } + \frac{(y-k) ^{2} }{ b^{2} } = 1[/tex]

where variables a and b and the two different measurements of the vertices

The equation of a hyperbola is: 

[tex] \frac{((x-h)^{2} }{ a^{2} } - \frac{(y-k)^{2} }{ b^{2} } = 1[/tex] if it is with a horizontal transverse axis

[tex] \frac{ (y-k)^{2} }{ b^{2} } - \frac{ (x-h)^{2} }{ a^{2} } = 1[/tex] if it is with a vertical transverse axis

Notice these have a subtraction operation, the exact opposite ellipse.