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Rotated Conic: $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$ , $B \neq 0$

When B is not zero, the conic section may be a rotated ellipse, hyperbola, or parabola. The angle of rotation can be found by the transformation of axes.

Ellipse (Horizontal): $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , $a > b$

Horizontal ellipse: the longer axis (major axis) is horizontal. Foci are located at (h±c, k), where $c^2 = a^2 - b^2$ .

Ellipse Eccentricity: $e = \sqrt{1 - \frac{b^2}{a^2}}$ , $a > b$

The eccentricity of an ellipse measures the deviation of the ellipse from being a circle. Eccentricity closer to 0 means the shape is more circular, closer to 1 means more elongated.

Hyperbola Eccentricity: $e = \sqrt{1 + \frac{b^2}{a^2}}$

The eccentricity of a hyperbola measures the extent of the 'openness' of the hyperbola. For hyperbolas, the eccentricity is always greater than 1.

Parabola (Horizontal): $(y-k)^2 = 4p(x-h)$

In this form, the parabola opens to the right if p > 0 and to the left if p < 0. The focus is at (h+p, k) and the directrix is the line x = h - p.

Parabola (Vertical): $(x-h)^2 = 4p(y-k)$

In this form, the parabola opens upward if p > 0 and downward if p < 0. The focus is at (h, k+p) and the directrix is the line y = k - p.

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

Vertical hyperbola: the transverse axis is vertical. Foci are located at (h, k±c), where $c^2 = a^2 + b^2$ .

Rectangular Hyperbola: $xy=c^2$

A rectangular hyperbola is a hyperbola with asymptotes that are perpendicular to each other. Its eccentricity is $\sqrt{2}$ , and it has two branches that are symmetrical about the origin.

Parabola: $y=ax^2+bx+c$

A parabola is the set of all points equidistant from a point, the focus, and a line, the directrix. The vertex is the minimum or maximum point of the parabola.

Ellipse (Vertical): $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , $a < b$

Vertical ellipse: the longer axis (major axis) is vertical. Foci are located at (h, k±c), where $c^2 = b^2 - a^2$ .

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

Horizontal hyperbola: the transverse axis is horizontal. Foci are located at (h±c, k), where $c^2 = a^2 + b^2$ .

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

An ellipse has two focal points. The sum of the distances from any point on the ellipse to the focal points is constant. Major axis length is 2a, minor axis is 2b.

Degenerate Conic: $(x-h)^2 + \lambda(y-k)^2 = 0$

A degenerate conic can result in a single point, a pair of intersecting lines, or no graph at all. Represents collapsed forms of ellipses, hyperbolas, or parabolas.

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

A hyperbola consists of two separate curves called branches. The difference of distances from any point on the hyperbola to the foci is constant. Asymptotes are lines that the hyperbola approaches at infinity.

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

A set of points at a constant distance, r, from a central point (h,k). A circle is a special case of an ellipse where the major and minor axes are equal.

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