Conic Sections

What just happened?

You tilted the cutting plane so that it is parallel to the slant line of the cone. This one special angle closes the family of ellipses and opens the curve — you get a parabola. The type of cross-section depends only on the cut angle, not on how large the cone is.

What are conic sections?

Conic sections (circle, ellipse, parabola, hyperbola) are the curves produced by cutting a cone with a plane at various angles. They were systematically studied by the ancient Greek mathematician Apollonius of Perga in his monumental eight-book work "Conics" (Conica, c. 200 BC) — it was he who gave these curves the names still used today: ellipse, parabola and hyperbola.

Four cuts of a single cone:

Cutting a double cone with a plane, depending on the angle $\theta$ relative to the slant-line angle, we obtain:

• a horizontal cut (perpendicular to the axis) → circle, $x^2+y^2=r^2$;
• a slightly oblique cut (angle smaller than the slant line) → ellipse, $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$;
• a cut parallel to the slant line → parabola, $y=ax^2$;
• a steep cut (crosses both halves) → hyperbola, $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$.

Equation of a circle:

— a circle with centre $(a,\,b)$ and radius $r$; for a centre at the origin it simplifies to $x^2+y^2=r^2$. It is the simplest of the conic sections and the starting point of analytic geometry.

The invariant — the angle alone decides:

The eccentricity $e$ depends only on the cut angle $\theta$ (and the fixed slant-line angle $\alpha$), not on the size of the cone: $e=0$ → circle, $0<e<1$ → ellipse, $e=1$ → parabola, $e>1$ → hyperbola.

The same curves govern the heavens:

More than 1800 years later it turned out that conic sections describe the motion of celestial bodies: Kepler discovered that planets orbit along ellipses (with the Sun at a focus), and the path of a thrown body (a projectile) in a uniform gravitational field is a parabola; a non-periodic comet races along a hyperbola. The parabola is also the basis of satellite dishes and reflectors — they focus rays at the focus.

Why you need this for the exam:

The circle $(x-a)^2+(y-b)^2=r^2$ and the parabola $y=ax^2+bx+c$ are the daily bread of analytic geometry and the quadratic function. Trick: recognise the equation of the curve and remember that one cone hides all four.

Remember: circle, ellipse, parabola and hyperbola are one family — cross-sections of a cone. Which one you get is decided by the cut angle alone: parallel to the slant line gives a parabola.