The Ellipse and Hyperbola Lab
Ellipses and hyperbolas: centered, shifted, and in general form
Step 1 of 9: Two Pushpins and a Loop of String
Drag the pencil point around, keeping the string taut between the two pins. Watch the shape that traces out.
You have made one. Now: what did the string enforce?
Step 2 of 9: The Constant Sum
The string length never changed. The string runs from pin to pencil to pin. So the sum of the distances to the two pins is constant. That is the definition of an ellipse.
Drag along the ellipse. The live readout confirms the sum holds at every point on the curve.
Step 3 of 9: The Semi-Major Axis
That constant sum has a name: \(2a\), where \(a\) is the semi-major axis. Measure the long axis of the ellipse. What is the relationship between the constant sum and the length of the major axis?
Step 4 of 9: Eccentricity
Drag the pins together and apart. Watch what happens to the ellipse shape.
A circle is an ellipse whose foci coincide. Eccentricity \(e = c/a\): zero when the pins touch, approaching 1 as they separate.
Step 5 of 9: Standard Form
Place the ellipse centered at the origin. The distance definition says: for every point \((x, y)\) on the ellipse, the sum of distances to the two foci equals \(2a\).
Starting from the distance definition with foci at \((\pm c, 0)\):
\(\sqrt{(x+c)^2 + y^2} + \sqrt{(x-c)^2 + y^2} = 2a\)
Which denominator is bigger tells you which way the ellipse is long. That is the whole rule.
Step 6 of 9: Off-Center
Drag the whole ellipse away from the origin. Watch the equation update.
It is a translation. Nothing else changed. The equation becomes \(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\).
Step 7 of 9: Not in Standard Form
\(4x^2 + 9y^2 - 16x + 54y + 61 = 0\). Is that an ellipse? You cannot tell. Complete the square on both variables to find out.
Group x and y terms:
\(4(x^2 - 4x) + 9(y^2 + 6y) = -61\)
Step 8 of 9: The Hyperbola -- Same Pins, New Rule
Same two pins. But now, instead of the sum, the DIFFERENCE of distances stays constant. Drag a point that maintains a constant difference. Two branches appear.
One sign flip in the definition. A completely different shape.
Step 9 of 9: The Aha -- Sum vs. Difference
Ellipse and hyperbola are the same construction with one operator changed: sum versus difference. Every feature of both traces back to that one choice.
The standard form for a hyperbola centered at the origin with a horizontal transverse axis is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). The asymptotes are \(y = \pm\frac{b}{a}x\). The relationship is \(b^2 = c^2 - a^2\). Off-center: \(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\).
Explore Ellipses and Hyperbolas
Adjust the sliders to see how each parameter changes the shape. Toggle between ellipse and hyperbola to compare.
Completing the Square Workbench
Enter a general-form conic equation coefficients: \(Ax^2 + Cy^2 + Dx + Ey + F = 0\)
Try This
Earth's Orbit
Earth's orbit has a perihelion (closest to Sun) of 91.4 million miles and an aphelion (farthest from Sun) of 94.5 million miles. Find \(a\), \(c\), and the eccentricity. Is Earth's orbit closer to a circle or to a stretched ellipse?
Hint: perihelion = a - c, aphelion = a + c. Tolerance: within 0.1 million miles for a and c, within 0.01 for eccentricity.
Whispering Gallery
A whispering gallery is 40 feet long and 25 feet wide. Where exactly should two people stand to hear each other's whispers? Give the coordinates of both foci relative to the center of the room.
The room is an ellipse with major axis 40 ft and minor axis 25 ft. Tolerance: within 0.5 feet.
LORAN Navigation
LORAN navigation put a ship on a hyperbola by timing signals from two stations. Two stations sit 300 miles apart. Timing says you are 100 miles closer to Station A than to Station B.
Write the equation of the hyperbola you are on (place stations at the foci on the x-axis, centered at the origin).
The constant difference |d1 - d2| = 100, and the stations are 300 miles apart so c = 150. Tolerance: within 1 for a and b values.
- Define an ellipse as the set of points with a constant sum of distances to two foci
- Identify and graph ellipses centered at the origin and not at the origin
- Convert general-form equations to standard form by completing the square
- Define a hyperbola as the set of points with a constant difference of distances to two foci
- Identify and graph hyperbolas with their asymptotes, vertices, and foci
- Compare ellipse and hyperbola as two outcomes of the same geometric construction
- Apply ellipse and hyperbola equations to real-world scenarios
Quick Check
An ellipse has \(a = 5\) and \(b = 3\). What is \(c\), the distance from the center to each focus?
Instructor Notes
Teaching Notes
- The string construction in Steps 1-2 is the single most powerful entry point for ellipses. If students physically do this (even virtually), the "constant sum" definition stops being abstract.
- Step 4 (eccentricity) should land before the algebra. Once students see a circle morph into a flattened ellipse by moving the foci, \(e = c/a\) is just naming what they watched.
- Step 8 is the pivot: same pins, new rule. The surprise that switching from sum to difference produces a completely different curve is the conceptual anchor for the entire section.
- The completing-the-square steps (7, and the Explore workbench) are where students typically lose the thread. Let them use the guided steps rather than doing it from scratch.
Common Student Errors
- "a is always under x." It is not. The larger denominator tells you the direction of the major axis, regardless of whether it sits under x or y.
- Using \(b^2 = a^2 - c^2\) for hyperbolas. For hyperbolas it is \(b^2 = c^2 - a^2\). This is the single most common computational error in this chapter.
- Forgetting to multiply the completed-square constant by the leading coefficient. \(4(x^2 - 4x + 4)\) means adding 16 to the right side, not 4.
- Mixing up vertices and co-vertices. Vertices sit on the major axis (distance a from center), co-vertices on the minor axis (distance b).
Discussion Questions
- Why does changing sum to difference produce two branches instead of one closed curve? What does the algebra force?
- Why is eccentricity always between 0 and 1 for an ellipse? What would e = 1 look like?
- In the whispering gallery, why does any sound from one focus reach the other focus, no matter the direction?
- LORAN uses hyperbolas. GPS uses circles (spheres in 3D). What is the mathematical difference in the approach?
Exam Connection
- Standard exam tasks: convert general form to standard form, identify center/vertices/foci/eccentricity, sketch.
- The \(b^2 = a^2 - c^2\) vs. \(b^2 = c^2 - a^2\) distinction is tested directly on nearly every exam.
- Expect at least one "which denominator is bigger" orientation question.
- Applied problems (orbit, whispering gallery) appear as bonus or extra-credit questions.