'a' determines the direction and width. Positive 'a' opens right/up, negative opens left/down.

If a = b in an ellipse, it becomes a circle.

The term with the positive sign indicates the direction of opening.

It represents the center point of the conic section, whether it's a circle, ellipse, or hyperbola.

A larger distance between the center and foci results in a more elongated ellipse.

The foci of a hyperbola determine its curvature; as foci move further from the center, the hyperbola becomes wider.

Every point on the parabola is equidistant from the focus and the directrix.

They serve as guidelines for the branches of the hyperbola, indicating how the hyperbola extends to infinity.

Changing 'a' and 'b' alters the lengths of the major and minor axes, thus changing the ellipse's elongation and orientation.

Changing 'a' and 'b' alters the slopes of the asymptotes, thus changing the hyperbola's width and orientation.

Ellipse: '+' sign between squared terms, closed curve. | Hyperbola: '-' sign, two separate curves.

Circle: $(x-h)^2 + (y-k)^2 = r^2$, equal radii. | Ellipse: $\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$, different horizontal and vertical radii.

Parabola: One focus and directrix, one squared term. | Hyperbola: Two foci, two squared terms with a minus sign.

Ellipse: Foci are inside the curve, sum of distances to foci is constant. | Hyperbola: Foci are outside the curve, difference of distances to foci is constant.

Ellipse: 'a' is the semi-major axis length. | Hyperbola: 'a' is the distance from the center to the vertices along the transverse axis.

Ellipse: $c^2 = a^2 - b^2$. | Hyperbola: $c^2 = a^2 + b^2$.

Circle: The center is equidistant from all points on the circle. | Hyperbola: The center is the midpoint between the vertices and foci, but not on the hyperbola itself.

Hyperbola: Has asymptotes that guide the branches of the curve. | Ellipse: Does not have asymptotes.

Parabola: Determined by which variable is squared and the sign of the leading coefficient. | Hyperbola: Determined by which term (x or y) is positive in the standard equation.

Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. | Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ or $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$.

$(y - k)^2 = a(x - h)$

$a(y - k) = (x - h)^2$

$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

$(x-h)^2 + (y-k)^2 = r^2$

$\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$

$-\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$

$c^2 = a^2 - b^2$

$c^2 = a^2 + b^2$

$e = \frac{c}{a}$

$e = \frac{c}{a}$