Conic Sections and Parametric Equations

A parabola is the set of points equidistant from a fixed focus and a fixed line, the directrix. The number $p$ is the signed distance from vertex to focus.

• Vertical axis: focus $(h,\,k+p)$, directrix $y=k-p$. Opens up if $p>0$, down if $p<0$.
• Horizontal axis: focus $(h+p,\,k)$, directrix $x=h-p$. Opens right if $p>0$, left if $p<0$.
• Latus rectum (focal chord through the focus, perpendicular to the axis) has length $|4p|$.
• Eccentricity of every parabola is $e=1$.

• $a$ = semi-major axis (largest denominator, $a^2$); $b$ = semi-minor axis.
• Vertices are $a$ from center along the major axis; co-vertices are $b$ from center along the minor axis.
• Foci lie on the major axis, $c$ from center, where $c^2=a^2-b^2$.
• Eccentricity $e=\dfrac{c}{a}$, with $0<e<1$ (nearer $0$ is more circular).

• The positive term names the transverse axis; $a^2$ is its denominator (it need not be the larger one).
• Vertices are $a$ from center along the transverse axis; foci are $c$ from center with $c^2=a^2+b^2$.
• Asymptotes: $y-k=\pm\dfrac{b}{a}(x-h)$ for the left/right form; $y-k=\pm\dfrac{a}{b}(x-h)$ for the up/down form.
• Eccentricity $e=\dfrac{c}{a}$, with $e>1$.

With no $xy$ term, $Ax^2+Cy^2+Dx+Ey+F=0$ classifies by comparing $A$ and $C$:

• Circle: $A=C$ (and same sign), e.g. $A=C\ne0$.
• Parabola: exactly one of $A,C$ is $0$ (only one variable is squared).
• Ellipse: $A$ and $C$ have the same sign but $A\ne C$.
• Hyperbola: $A$ and $C$ have opposite signs ($AC<0$).

A parametric curve gives $x$ and $y$ each as a function of a third variable $t$:

As $t$ increases the point $(x,y)$ traces the curve in a definite direction (its orientation).

• Graph by a table: choose values of $t$, compute $(x,y)$, and connect in order of increasing $t$.
• Eliminate the parameter: solve one equation for $t$ (or use a trig identity) and substitute to get a relation in $x$ and $y$.

With a focus at the pole, a conic has polar equation

where $e$ is the eccentricity and $d>0$ is the distance from focus to directrix.

• $e=1\Rightarrow$ parabola,   $0<e<1\Rightarrow$ ellipse,   $e>1\Rightarrow$ hyperbola.
• $\cos$ means the directrix is vertical; $\sin$ means it is horizontal.

Take a DOUBLE cone (two ice-cream cones tip to tip, extending forever) and cut it with a plane. The angle of the cut decides everything: a horizontal cut gives a circle; tilt the plane a little and the circle stretches into an ellipse.

Tilt the plane until it is exactly PARALLEL to the cone's slant side and the curve never closes up — that single borderline angle produces the parabola. It is the knife-edge between the closed curves (ellipses) and the doubly-open one (hyperbola).

Tilt past the parallel position (or cut straight down) and the plane slices BOTH halves of the double cone, producing the two branches of a hyperbola. The two branches are one curve — the same plane meeting the same cone, twice.

Standard form $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b$): the long semi-axis is $a$, the short one $b$, and the two foci sit at $(\pm c, 0)$ with $c^2 = a^2 - b^2$. The defining property: every point of the ellipse has distances to the two foci summing to $2a$ (a loop of string around two pins traces one).

$x^2 = 4py$: focus $(0, p)$, directrix $y = -p$, and every point of the curve is EQUIDISTANT from the focus and the directrix. Satellite dishes are parabolas because incoming parallel rays all bounce to the focus.

$\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ opens left-right with foci at $(\pm c, 0)$, $c^2 = a^2 + b^2$ (note the PLUS — foci sit outside), and asymptotes $y = \pm\tfrac{b}{a}x$: the branches hug these lines forever without touching. Defining property: the DIFFERENCE of the focal distances is constant ($2a$).

Substitute a line $y=mx+c$ into a conic and collect a quadratic in $x$. The line is tangent exactly when that quadratic has a double root, i.e. its discriminant is $0$ (secant if the discriminant is positive, misses if negative). Carrying this out once gives memorable tangency conditions for lines of slope $m$:

• Ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$:  tangent iff $c^2=a^2m^2+b^2$.
• Hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$:  tangent iff $c^2=a^2m^2-b^2$.
• Parabola $y^2=4px$:  tangent iff $c=\dfrac{p}{m}$ (line $y=mx+\tfrac{p}{m}$).

These same relations let you find a conic from tangency, point, or focus data: each condition gives one equation in the unknowns $a^2,b^2,p,\dots$, and you solve the resulting system.

A focal chord is any chord through a focus. The latus rectum is the focal chord perpendicular to the major/transverse axis; its half-length $\ell$ (the semi-latus rectum) controls the polar form below.

For an ellipse $\ell=a(1-e^2)$ and for a hyperbola $\ell=a(e^2-1)$, so knowing $a$ and $e$ recovers the full shape. Every focal chord of a conic has semi-latus rectum $\ell$ as its harmonic-mean radius.

The full second-degree equation

may include a cross term $Bxy$, which rotates the axes. The rotation-invariant discriminant $B^2-4AC$ classifies the curve without ever finding the rotation angle:

• $B^2-4AC<0\Rightarrow$ ellipse (circle if also $A=C,\ B=0$),
• $B^2-4AC=0\Rightarrow$ parabola,
• $B^2-4AC>0\Rightarrow$ hyperbola.

When $B=0$ this reduces to the earlier sign test on $A$ and $C$. Shortest distance to a conic: the closest point of a smooth conic to an external line lies where a tangent is parallel to that line, so slide the tangency condition to the tangent line nearest the given one, then measure the gap between the two parallel lines.