Spatial Geometry & Functions · AP Precalculus Unit 4 (Conic Sections & Polar Coordinates), A-Level Further Maths (Conics), High School Analytic Geometry.
Conic Sections & Ellipse Construction Visualizer
Why does a planar cone slice produce a curve with constant focal distance sum? Experience Dandelin's brilliant spheres in 3D. Pull a virtual string across two pins, adjusting eccentricity e and focal spacing.
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Live Derivations & Metrics
semi_b
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b = √(a² - c²)
eccentricity_e
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e = c / a (0 ≤ e < 1)
string_len_2a
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Invariant PF1 + PF2 = 2a
pin_dist_2c
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|F1F2| = 2c
Real-Time Formula Derivation
Governing Formula:
--Substitution:
--Regime: --
Worked Examples & Exam Question Deep Dive
Analytic Geometry: Ellipse with a=5, c=4, Find Standard Equation & Directrix
Problem: An ellipse has foci on the x-axis with semi-major axis a = 5 and focal half-distance c = 4. Find its standard equation, eccentricity e, and directrix equations.
Analytical Strategy: From a² = b² + c², we get b² = 25 - 16 = 9 (b=3). Standard equation is x²/25 + y²/9 = 1. Eccentricity e = c/a = 4/5 = 0.8. Directrices are x = ±a²/c = ±25/4.
Step-by-Step Derivation:
- Step 1: Compute semi-minor axis b: b = √(a² - c²) = √(25 - 16) = 3.
- Step 2: Foci on x-axis => equation is x²/25 + y²/9 = 1.
- Step 3: Compute eccentricity e = c / a = 4 / 5 = 0.8.
- Step 4: Compute directrices: x = ±a²/c = ±25/4 = ±6.25.
⚠️ Common Misconception Pitfall: Confusing conic relationships: for an ellipse, a is largest (a² = b² + c²); for a hyperbola, c is largest (c² = a² + b²).
Student Inquiry Worksheet & Practice Problems
Q1. Two pins are spaced 6 cm apart holding a 10 cm string taut. What is the semi-minor axis b?
Answer Key: 4 cm (2a=10=>a=5, 2c=6=>c=3, b=√(5²-3²)=4)
Q2. Which physical property of an ellipse is utilized in medical lithotripters for kidney stone treatment?
Answer Key: Acoustic reflection focal property (waves emitted from one focus reflect and concentrate at the other focus)
Frequently Asked Questions & Misconceptions
How do slice angle β and cone generator angle α dictate the conic section type?
Comparing slice angle β to cone generator slope α: β=0 gives a circle; β < α gives an ellipse; β = α (parallel to generator) produces a parabola; β > α (cutting both nappes) produces a hyperbola.