Summary

Conic sections (circle, ellipse, parabola, hyperbola) all arise from slicing a double right circular cone at different angles $\beta$ against its semi-vertical angle $\alpha$: $\beta=90^\circ$ gives a circle, $\alpha<\beta<90^\circ$ an ellipse, $\beta=\alpha$ a parabola, $0\le\beta<\alpha$ a hyperbola; a plane through the vertex instead gives the degenerate cases -- a point, a line, or a pair of intersecting lines.

Circle: locus of points at fixed distance $r$ from a centre $(h,k)$.

Centre $(-g,-f)$, radius $\sqrt{g^2+f^2-c}$.

Parabola ($y^2=4ax$, $a>0$): locus equidistant from focus $(a,0)$ and directrix $x=-a$. Vertex $(0,0)$, latus rectum $4a$, eccentricity $e=1$ always. Three further orientations: $y^2=-4ax$, $x^2=4ay$, $x^2=-4ay$.

Ellipse ($\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b>0$): locus with $PF_1+PF_2=2a$. Foci $(\pm c,0)$, $b^2=a^2-c^2=a^2(1-e^2)$, $e=c/a\in(0,1)$, vertices $(\pm a,0)$, latus rectum $2b^2/a$. …