9. Conics and quadrics through equations and sections

Learning goals and definitions

Complete squares, classify basic conics and quadrics, and use sections to interpret higher-dimensional equations. A conic is a plane locus defined by a nonzero quadratic equation; a quadric in three dimensions is defined analogously. Equations may describe degenerate or empty sets. A complete treatment of arbitrary rotated forms uses linear algebra; here we derive translations and one planar rotation explicitly.

For positive $a,b$, the ellipse $x^2/a^2+y^2/b^2=1$ is bounded. The hyperbola $x^2/a^2-y^2/b^2=1$ has two unbounded branches, and the parabola $y=x^2$ has a single axis and vertex. Their translated versions replace $x,y$ by $x-h,y-k$. Completing squares gives these normal forms without altering the locus.

Proposition: a focus–directrix parabola

The points equidistant from the focus $(0,p)$ and directrix $y=-p$, where $p>0$, satisfy $x^2=4py$.

Proof. The squared distance to the focus is $x^2+(y-p)^2$ and squared distance to the directrix is $(y+p)^2$. Equating and simplifying gives $x^2=4py$. Conversely that equation implies equality of the nonnegative squared distances, hence equality of distances. It also forces $y\ge0$, consistent with the geometry. Squaring here introduces no extraneous points because both distances are nonnegative.

For a centered planar form $Ax^2+Bxy+Cy^2$, a rotation

$$x=u\cos\theta-v\sin\theta,\qquad y=u\sin\theta+v\cos\theta$$

has transformed mixed coefficient $(C-A)\sin2\theta+B\cos2\theta$, by direct expansion. One can choose an angle to make that coefficient zero. Translation then removes linear terms in any direction with nonzero quadratic coefficient. Zero coefficients must be handled separately; completing a square by dividing by zero is not legitimate.

Worked examples

Translation and degeneracy. The equation $x^2+4y^2-2x+8y+1=0$ becomes $(x-1)^2+4(y+1)^2=4$, or $(x-1)^2/4+(y+1)^2=1$. It is an ellipse centered at $(1,-1)$ with semiaxes $2$ and $1$. If the right side were zero, the locus would instead be the single point $(1,-1)$; if negative, it would be empty.

A rotated conic. For $xy=1$, choose $x=(u-v)/\sqrt2$, $y=(u+v)/\sqrt2$. Then $xy=(u^2-v^2)/2$, so the equation becomes $u^2-v^2=2$, a hyperbola in rotated axes. The coordinate substitution is invertible, so it neither loses nor adds points.

Sections of quadrics. Let $a,b,c>0$. The ellipsoid $x^2/a^2+y^2/b^2+z^2/c^2=1$ is bounded; a section at fixed $z$ is an ellipse when $|z|<c$, a point when $|z|=c$, and empty beyond. The surface $x^2+y^2-z^2=1$ has circular sections of radius $\sqrt{1+z^2}$ for every $z$, a one-sheet hyperboloid. The equation $z^2-x^2-y^2=1$ has no points when $|z|<1$ and two separated sheets. The cone $z^2=x^2+y^2$ has a singular vertex where its two nappes meet.

Graded practice

1. Foundation. Complete squares and identify $x^2+y^2+4x-6y+9=0$.
2. Core. Find the focus and directrix of $x^2=12y$.
3. Core. Describe sections of $z=x^2+y^2$ at $z=k$, at $x=0$, and at $y=0$.
4. Proof. Show that $x^2+y^2+z^2=0$ is a single point over $\mathbb R$, while $x^2+y^2-z^2=0$ is not. Why is the sign of a quadratic form essential?

Full solutions

1. The equation is $(x+2)^2+(y-3)^2=4$. It is a circle centered at $(-2,3)$ with radius two, a special ellipse with equal semiaxes.
2. Compare with $x^2=4py$ to obtain $p=3$. The focus is $(0,3)$ and directrix is $y=-3$.
3. If $k>0$, the horizontal section is a circle of radius $\sqrt k$; if $k=0$, it is the vertex; if $k<0$, it is empty. The sections $x=0$ and $y=0$ are the parabolas $z=y^2$ and $z=x^2$, respectively. Together these support the name elliptic paraboloid, though sections are not a substitute for its defining equation.
4. Every square is nonnegative, so their sum is zero only when all three coordinates vanish. With the minus sign, $(t,0,t)$ satisfies the second equation for every real $t$, so it has infinitely many points and is a cone. Opposite signs allow nonzero terms to cancel; positive definiteness prevents such cancellation.

Summary and term definitions

A conic is a quadratic plane locus, and a quadric its higher-dimensional counterpart. A section intersects a surface with a specified plane. A degenerate locus has a lower-dimensional or otherwise collapsed form, such as a point or crossing lines. Translation shifts centers; orthogonal rotation changes axes without changing distances. State positivity assumptions on semiaxis parameters and check whether the resulting equation is empty before assigning a geometric label.