Twelfth Grade · Math · week 35

Conic sections: ellipses and hyperbolas

Write and interpret the standard-form equation of an ellipse or hyperbola, finding its center, vertices, foci, and (for a hyperbola) its asymptotes.

An ellipse is the set of points whose distances to two fixed points (the foci) always add up to the same total. Its standard form is x^2/a^2 + y^2/b^2 = 1, centered at the origin, where a^2 is always paired with the LARGER denominator -- that tells you whether the major axis is horizontal or vertical. The relationship c^2=a^2-b^2 locates the foci a distance c from center along the major axis, and eccentricity e=c/a (always between 0 and 1) measures how stretched the ellipse is: closer to 0 is nearly circular, closer to 1 is very elongated.

A hyperbola is the set of points whose distances to two foci always have the same DIFFERENCE (instead of sum). Its standard form is x^2/a^2 - y^2/b^2 = 1 (opening left-right) or y^2/a^2 - x^2/b^2 = 1 (opening up-down), with c^2=a^2+b^2 this time (addition, not subtraction) and eccentricity e=c/a always greater than 1. A hyperbola also has two asymptote lines that the curve approaches but never touches, with slopes +-(b/a) for a horizontal transverse axis or +-(a/b) for a vertical one.

📋 For the teacher
Teach it (10–15 min):
  • Drill "a^2 goes with the LARGER denominator" as a standalone rule for ellipses, separate from any assumption about which variable it attaches to.
  • Contrast c^2=a^2-b^2 (ellipse, subtraction) against c^2=a^2+b^2 (hyperbola, addition) side by side, since this is the single easiest fact to mix up between the two conics.
  • Sketch the asymptotes of a hyperbola FIRST, as two crossing lines through the center, before attempting to sketch the curve itself -- the curve hugs those lines by definition.
  • Connect eccentricity's two ranges (0 to 1 for an ellipse, greater than 1 for a hyperbola) to a mental picture: a stretched-out ellipse approaches an eccentricity of 1 right where a hyperbola's range begins.
Talk about it:
  • Why does a^2 pair with the larger denominator in an ellipse, regardless of which variable that denominator is under?
  • Why does an ellipse use c^2=a^2-b^2 while a hyperbola uses c^2=a^2+b^2?
  • Why must a hyperbola's eccentricity be greater than 1, while an ellipse's must be less than 1?

Mastery looks like: They correctly identify a, b, c, the foci, and (for a hyperbola) the asymptotes from a standard-form ellipse or hyperbola equation, on either a horizontal or vertical axis.

Common stumbles: Assuming a^2 always goes with x^2; confusing the ellipse's subtraction relationship (c^2=a^2-b^2) with the hyperbola's addition relationship (c^2=a^2+b^2).

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Records attendance + 15 min of Math for this learner.
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