Conic Curves Slides Conic Curves
Ellipses & Hyperbolas
Definitions
Construction
Equations
The Ellipse Definition
An ellipse is the set of all points where the SUM of the distances to two fixed points (foci) is constant.
Think about the String:
The string length stays the same.
Distance 1 + Distance 2 = Constant
\(d_1 + d_2 = \text{string length}\)
The Hyperbola Definition
A hyperbola is the set of all points where the absolute DIFFERENCE of the distances to the foci is constant.
Key Distinction:
One distance gets longer as the other gets longer.
\(|d_1 - d_2| = \text{Constant}\)
\(|d_1 - d_2| = 2a\)
The Distance Foundation
To derive the equations, we start with the Distance Formula between a generic point \((x, y)\) and a focus \((c, 0)\).
Ellipse Setup
\(d_1 + d_2 = 2a\)
\(\sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a\)
We isolate one radical and square both sides... twice!
Hyperbola Setup
\(|d_1 - d_2| = 2a\)
\(\sqrt{(x+c)^2 + y^2} - \sqrt{(x-c)^2 + y^2} = \pm 2a\)
The process is similar, but the sign change leads to a minus in the final form.
Decoding the Variables
a
Distance to Vertex
Half the length of the major axis.
c
Distance to Focus
Distance from center to a focus.
b
The Bridge Variable
Created to simplify the algebra.
Pythagorean Relationships:
Ellipse: \(a^2 = b^2 + c^2\)
Hyperbola: \(c^2 = a^2 + b^2\)
"Follow the focus! In a hyperbola, the focus is the furthest out (\(c > a\))."
String and Pins Lab String and Pins Lab
Exploring the Geometric Definitions of Conics
Name:
Date:
2 Push Pins
Cardboard Backing
String (20cm)
Ruler
1
The Ellipse: Fixed Sum
Procedure:
Place two pins 10cm apart on the cardboard. These are your foci .
Tie your 20cm string into a loop (or tie ends to the pins).
Place your pencil inside the loop, pull it taut, and draw the curve.
Observation:
As you move the pencil, does the total length of the string change?
Sketch your ellipse here
Connecting to the Definition
Let focus 1 be \(F_1\) and focus 2 be \(F_2\). Let the point on the ellipse be \(P\).
Definition:
\(PF_1 + PF_2 =\) Constant
1. Identify the Points
Generic Point: \(P(x, y)\)
Foci: \(F_1(-c, 0)\) and \(F_2(c, 0)\)
2. Apply Distance Formula
Write the formula here...
2
The Hyperbola: Fixed Difference
Constructing a hyperbola with string is trickier. Instead of a loop, we look at the difference between distances. Imagine two people holding ends of strings of different lengths that both connect to the same moving pencil.
\(|d_1 - d_2| = 2a\)
Predict the Curve Shape:
A. Circular
B. Two Branches
C. Single Arch
Thinking Deeply:
If the distance between foci is 10, and the constant difference is 6, what is the furthest a point on the curve can be from the center? Justify using your string mental-model.
Daily Mastery Goal:
I can explain how foci determine the shape of a conic curve.
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2
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5
Equation Builder Worksheet Equation Builder
Step-by-Step Ellipse Derivation
ID: MATH-GEO-GPE.3
Name:
The Goal: We want to turn the sentence "The sum of the distances is constant" into the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). This worksheet scaffolds the heavy algebra into small bites.
Step 01
Set the Equation
Using the distance formula for point \((x,y)\) and foci \((\pm c, 0)\), we start with:
\(\sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a\)
Task: Isolate the first radical by subtracting the second radical from both sides.
\(\sqrt{(x+c)^2 + y^2} = \) _________________________________________________
Step 02
The First Square
When we square both sides, we get a bit of a mess! We have simplified it for you below. Cross out the terms that appear on both sides of the equals sign.
\[x^2 + 2xc + c^2 + y^2 = 4a^2 - 4a\sqrt{(x-c)^2 + y^2} + x^2 - 2xc + c^2 + y^2\]
Hint: Look for \(x^2\), \(c^2\), and \(y^2\).
Step 03
Move and Divide
After crossing things out and moving \(2xc\), we get: \(4xc - 4a^2 = -4a\sqrt{(x-c)^2 + y^2}\).
Write your simplified version here...
Step 04
The "B" Substitution
After squaring again and rearranging, we reach a point where we have a term \((a^2 - c^2)\). In an ellipse, we know that the distance to the vertex \(a\) is greater than the distance to the focus \(c\).
We define a new letter: \(b^2 = a^2 - c^2\).
Why do we use \(b\)?
It makes the equation look cleaner.
It represents the minor axis height.
It connects to the Pythagorean Theorem.
Final Standard Form
\[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\]
Quick Check:
If an ellipse has foci at \((\pm 4, 0)\) and the constant sum is \(2a = 10\)...
Find \(a\):
Find \(c\):
Calculate \(b^2\) using \(a^2 - c^2\):
Write the full equation:
Curve Monitoring Teacher Guide Curve Monitoring
Tier 2 Intervention Tracker & Exit Ticket
Target: HS.G-GPE.A.3
Small Group Observation Checklist
Student Name Defines Conic by Distance Sum/Diff Identifies Foci on Construction Relates \(a, b, c\) Algebraically Notes / Misconceptions 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
Focus Check
Name: ______________________
1. Circle the correct word to complete the definition:
An ellipse is the set of all points where the SUM / DIFFERENCE of the distances from the foci is constant.
2. Match the geometric feature to its algebraic letter:
Distance from center to Focus __
Distance from center to Vertex __
a c
3. Reflection: If we move the two pins (foci) further apart, what happens to the shape of the ellipse? (Use your experience from the lab!)
Proficient (3/3)
Moves to independent practice.
Approaching (2/3)
Peer tutoring on variable definitions.
Needs Support (0-1/3)
Retest using interactive desmos tool.
