Conic Sections

A conic section is a curve you get by slicing a double cone with a flat plane. Tilting the plane a little more each time produces, in order, a circle, an ellipse, a parabola, and a hyperbola. Every one of them can be written as a single second-degree equation

and the whole topic is really one question asked four ways: what shape does this equation draw, and where are its key points?

For points $(x_1,y_1)$ and $(x_2,y_2)$:

The distance formula is the circle equation in disguise (a circle is all points a fixed distance from the center), so keep it close.

A circle with center $(h,k)$ and radius $r$ is the set of all points a distance $r$ from the center:

Read the center straight off the equation as the opposites of the numbers inside the parentheses, and take $r=\sqrt{\,r^2\,}$.

A parabola is every point equidistant from a fixed focus and a fixed line called the directrix. In vertex form:

Either way the vertex is $(h,k)$: it is the turning point, halfway between the focus and the directrix.

Write an upward parabola as $y=\dfrac{1}{4p}(x-h)^2+k$. Then $p$ is the distance from the vertex to the focus (and to the directrix):

The focus sits inside the curve; the directrix is the same distance out the other side.

The larger denominator sits under the axis holding the major axis (the long one). Let $a$ be the larger of the two, $b$ the smaller. Then:

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• Vertices: endpoints of the major axis, distance $a$ from center.
• Co-vertices: endpoints of the minor axis, distance $b$ from center.
• Foci: distance $c$ from center along the major axis, where $c^2=a^2-b^2$.

A shifted ellipse uses $\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1$ with center $(h,k)$.

The positive term tells you which way it opens; $a$ is always the number under the positive term.

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• Vertices: distance $a$ from center, on the opening axis.
• Foci: distance $c$ from center, where $c^2=a^2+b^2$ (note the $+$!).
• Asymptotes: for $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ they are $y=\pm\dfrac{b}{a}x$; for the up--down form they are $y=\pm\dfrac{a}{b}x$.

Given $Ax^2+Cy^2+Dx+Ey+F=0$ (no $xy$ term), look only at $A$ and $C$:

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• Circle: $A=C$ (same coefficient on $x^2$ and $y^2$).
• Parabola: exactly one of $A,C$ is $0$ (only one squared term).
• Ellipse: $A\ne C$ but $A$ and $C$ have the same sign.
• Hyperbola: $A$ and $C$ have opposite signs.

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• Ellipses --- orbits: planets and satellites travel in ellipses with the larger body at one focus (Kepler's First Law).
• Ellipses --- whispering galleries: a sound starting at one focus reflects off the elliptical ceiling and gathers at the other focus, so a whisper carries across the room.
• Parabolas --- dishes and headlights: a satellite dish or reflector collects incoming rays at its focus (and a bulb at the focus sends out a straight beam).
• Hyperbolas --- navigation and comets: some comets swing past the sun on an open hyperbolic path and never return.

So far you have read a conic off its equation. The reverse skill --- constructing the equation from geometric conditions --- is what real problems ask for. Each unknown coefficient needs one condition:

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• Through given points: plug each point into the general form and solve the resulting linear system for the coefficients.
• Given foci and eccentricity: the foci fix the center and $c$; then $e=\dfrac{c}{a}$ gives $a$, and $b$ comes from $b^2=a^2-c^2$ (ellipse) or $b^2=c^2-a^2$ (hyperbola).
• Given a tangent line: impose the tangency condition (next box) as one more equation.

Eccentricity $e$ measures how “stretched” a conic is: $e=0$ circle, $0<e<1$ ellipse, $e=1$ parabola, $e>1$ hyperbola.

To find where a line meets a conic, substitute the line into the conic. You get a quadratic in one variable, so the discriminant $\Delta=B^2-4AC$ of that quadratic tells you everything:

So “the line is tangent to the conic” translates into the single algebraic equation $\Delta=0$. For the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ this collapses to the tidy rule: the line $y=mx+k$ is tangent exactly when

A focal chord is any chord through a focus. The special focal chord perpendicular to the major (or focal) axis is the latus rectum, and its length measures how “wide” the curve is at the focus:

The reflection property explains every application from Section 7: a ray aimed at one focus of an ellipse bounces to the other focus; a ray parallel to a parabola's axis reflects through its single focus; and a ray toward one focus of a hyperbola reflects away along the line to the other focus.

When an $xy$ term is present the conic is rotated, and reading $A$ vs. $C$ is no longer enough. The general equation

is classified by the invariant $B^2-4AC$:

This quantity is unchanged by any rotation of the axes, which is why it works even when $B\ne0$. (When $B=0$ it agrees with the sign test of Section 6.)

A “conic” equation does not always draw a smooth curve. When the slicing plane passes through the apex of the cone you get a degenerate conic:

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• A single point --- e.g. $x^2+y^2=0$ is only $(0,0)$ (a “circle of radius $0$”).
• One line / two parallel lines --- e.g. $x^2=4$ gives $x=\pm2$ (a degenerate parabola).
• Two intersecting lines --- e.g. $x^2-y^2=0$ gives $y=\pm x$ (a degenerate hyperbola).
• Empty set (no points) --- e.g. $x^2+y^2=-1$ has no real solutions.

Watch for these after completing the square: a right-hand side of $0$ or a negative number is the tell.

The closest point $Q$ on a conic to an external point $P$ has a clean geometric signature: the segment $PQ$ is perpendicular to the tangent line at $Q$ (it lies along the normal). For a circle this makes the answer immediate --- the nearest point lies on the line through $P$ and the center:

where $C$ is the center and $r$ the radius. For other conics you either minimize the squared distance $d^2$ with calculus or impose the normal condition and solve.