Week 1 Discussion Notes

Table of Contents

• Sketching Axes

• Conic Sections

  • Ellipses
  • Parabolas
  • Hyperbola

Sketching Axes

This is a right and wrong way to sketch axes. For example,

You can check if your axes are correct using the right-hand rule. There are several variants of the right-hand rule, but here is how I do it: point the non-thumb fingers of your right hand in the direction of the $x$-axis, and curl them in the direction of the $y$-axis. Your thumb should point in the direction of the $z$-axis if you've drawn them correctly.

Conic Sections

Conic sections are called so because they come from cutting cones with a plane. Look at this GeoGebra graph) to see what I mean: when you tilt the plane (e.g., by sliding $a$, $b$, or $c$) and look at their intersections, you'll see that there can only be three possibilities:

Ellipses

When the plane is close to being flat, you'll get an ellipse:

Parabolas

When the plane is parallel with one of the halves of the cone, you get a parabola:

Hyperbola

Finally, when the plane intersects both halves of the cones, you get a hyperbola: