Summary

A conic section is the curve obtained by intersecting a double-napped right circular cone with a plane. The four types — circle, parabola, ellipse, hyperbola — depend on the angle of the cutting plane relative to the cone's axis.

Circle: Set of all points equidistant from a fixed centre $(h,k)$. Standard equation: $(x-h)^2 + (y-k)^2 = r^2$, where $r$ is the radius.

Parabola: Set of points equidistant from a fixed focus and a fixed directrix. Standard forms:

• $y^2 = 4ax$ (opens right), focus $(a,0)$, directrix $x = -a$, axis $y=0$, length of latus rectum $4a$.
• $x^2 = 4ay$ (opens up), focus $(0,a)$, directrix $y = -a$, axis $x=0$, latus rectum $4a$.
• For vertex at $(h,k)$, replace $x$ with $x-h$ and $y$ with $y-k$ in the standard equation.

Ellipse: Set of points where sum of distances to two foci is constant ($2a$). Standard equation (centre at origin, major axis along $x$-axis): $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a > b > 0$.

• Foci: $(\pm c,0)$, $c^2 = a^2 - b^2$.
• Vertices: $(\pm a,0)$; minor axis endpoints: $(0,\pm b)$.
• Eccentricity $e = \frac{c}{a}$ ($0 < e < 1$); length of latus rectum $\frac{2b^2}{a}$.

Hyperbola: Set of points where absolute difference of distances to two foci is constant ($2a$). Standard equation (centre at origin, transverse axis along $x$-axis): $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$.

• Foci: $(\pm c,0)$, $c^2 = a^2 + b^2$.
• Vertices: $(\pm a,0)$; asymptotes: $y = \pm \frac{b}{a}x$.
• Eccentricity $e = \frac{c}{a}$ ($e > 1$); length of latus rectum $\frac{2b^2}{a}$. …