Curve Clash Slides
CURVE CLASH
Ellipses vs. Hyperbolas
Objective
Compare and contrast the geometric properties, equations, and focal relationships of ellipses and hyperbolas.
Targets
- Identify \(c^2\) formulas for each conic.
- Explain \(a\) vs \(c\) spatial relationships.
Warm-up
5 MIN
On your Conic Comparison Worksheet:
"What do you already know?"
Label your Venn Diagram circles:
ELLIPSES and HYPERBOLAS.
Think: Equations? Shape? Real-world examples?
Intro Analysis
10 MIN
Watch 0:00 - 3:00. Focus on how the \(c^2\) formulas and standard equations differ.
Embedded media
Look For:
The sign between terms (\(+\) vs \(-\)).
Listen For:
How \(a, b,\) and \(c\) interact differently.
The Equation Clash
ELLIPSE
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
\[ c^2 = a^2 - b^2 \]
Distance to Focus < Vertex
HYPERBOLA
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
\[ c^2 = a^2 + b^2 \]
Distance to Focus > Vertex
Sorting Challenge
25 MIN
1
Sort your deck of Property Cards into the Venn Diagram.
2
Work with your partner to place formulas, graphs, and vocab.
Crucial Question: Why is \(c > a\) for hyperbolas but \(a > c\) for ellipses?
Classification
Left Only (Ellipse)
Overlap (Both)
Right Only (Hyperbola)
Final Verdict
5 MIN
Two Truths and a Lie
1. The value of \(c\) represents the distance from the center to a focus for both curves.
2. For a hyperbola, the vertices are always further from the center than the foci.
3. Both conics have standard equations equal to 1.
Spot the Lie!
Conic Comparison Worksheet
Conic Comparison
Pre-Calculus Unit 5
Curve Clash Activity
Name:
Date:
ELLIPSES
Hyperbolas
Shared Properties
Sort the property cards into these regions. Glue or tape once verified.
Critical Analysis
1. Compare the relationships between \(a\) and \(c\) for both conics. Why must \(c > a\) for a hyperbola, but \(a > c\) for an ellipse? Use the geometry of the curves in your explanation.
2. Looking at the \(c^2\) formulas (\(c^2 = a^2 - b^2\) vs. \(c^2 = a^2 + b^2\)), how does the sign in the standard equation relate to the sign in the focus formula?
Ref: YT-Iu-4-fizlD4 Pre-Calc Activity 04.2
Property Sorting Cards
Property Sorting Cards
Cut along the dashed lines. Sort into your Venn Diagram.
CURVE_CLASH_V1.0
Equation
\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
Equation
\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \]
Property
Sum of distances from any point to the foci is constant.
Focus Relationship
\[ c^2 = a^2 - b^2 \]
Focus Relationship
\[ c^2 = a^2 + b^2 \]
Property
Difference of distances from any point to the foci is constant.
Terminology
Major Axis
Terminology
Transverse Axis
Terminology
Vertices
Constraint
a > c
Constraint
c > a
Constraint
Center at (h, k)
Eccentricity
0 < e < 1
Eccentricity
e > 1
Feature
Asymptotes
Sketch
Sketch
Conic Contrast Teacher Guide
Teacher Guide
Curve Clash: Conic Comparison
Pre-Calculus / Geometry
Unit: Conic Sections
Lesson Pacing
0-5 MIN
Warm-up
Setup Venn Diagram
5-15 MIN
Video
Formula Intro
15-40 MIN
Activity
Property Sort
40-45 MIN
Closure
2 Truths & 1 Lie
Card Sorting Key
Ellipses Only
- \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
- \( c^2 = a^2 - b^2 \)
- Sum of distances... constant
- Major Axis
- \( a > c \)
- \( 0 < e < 1 \)
- Oval Sketch
Both (Shared)
- Vertices
- Center at (h, k)
- Foci (definitions)
Hyperbolas Only
- \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
- \( c^2 = a^2 + b^2 \)
- Diff. of distances... constant
- Transverse Axis
- \( c > a \)
- \( e > 1 \)
- Asymptotes
- Split Curve Sketch
Discussion Prompts
The "C" Dilemma: Why must \(c > a\) for hyperbolas? (Guide students to see that the foci must be 'outside' the vertices for the curve to wrap around them, whereas in an ellipse, the foci are 'inside' the boundary).
Equation Signs: Notice that the standard equation sign is the opposite of the sign in the focus formula. (Ellipse: + in equation, - in formula; Hyperbola: - in equation, + in formula).
Closure Key
The Lie:
"For a hyperbola, the vertices are always further from the center than the foci."
Reality: For a hyperbola, the Foci (\(c\)) are always further from the center than the vertices (\(a\)). Remember: \( c > a \) for hyperbolas.
