Calculate equations and properties of conic sections: circles, ellipses, parabolas, and hyperbolas. Fundamental curves in geometry and physics.
Last updated: April 2026 | By Patchworkr Team
Conic sections are curves obtained by intersecting a plane with a double cone. Depending on the angle of intersection, you get four types of curves: circle, ellipse, parabola, or hyperbola.
These curves appear throughout mathematics, physics, and engineering: planetary orbits (ellipses), projectile paths (parabolas), and navigation systems (hyperbolas).
Determine whether you're working with a circle, ellipse, parabola, or hyperbola based on the context or given information. Why: Each conic type has a fundamentally different equation form and parameter set. Knowing the type immediately tells you which equation template to use and which measurements are relevant.
Find or verify the center (h, k) for circles, ellipses, and hyperbolas; or the vertex (h, k) for parabolas. Why: The center/vertex is the reference point for the standard form. Everything else (radii, axes, focal parameters) is measured from this point. Getting the center wrong shifts the entire curve incorrectly.
Measure or calculate:
Why: Standard form is universally recognized and immediately reveals key properties. It allows you to read off the center, compute focal distances, eccentricity, and other derived quantities. Standard form is also required for computer graphing and analytical problem-solving.
Once the equation is established, calculate supporting values: for ellipses and hyperbolas compute eccentricity (e = c/a), focal distance (c), and asymptotes (for hyperbolas). For parabolas, find focus and directrix. Verify that all values are geometrically consistent. Why: Derived properties provide additional insight into the conic's behavior. Eccentricity reveals how far from circular a shape is. Focus and directrix positions are essential for reflective properties and physical applications (satellite dishes, telescope mirrors). Verification catches errors early.
Planetary Orbit (Ellipse)
Circle (e=0), ellipse (0<e<1), parabola (e=1), and hyperbola (e>1). The eccentricity (e) determines the shape.
By slicing a double cone with a plane at different angles. Horizontal gives a circle, tilted gives an ellipse, parallel to the side gives a parabola, and steep cuts both cones for a hyperbola.
A measure of how 'stretched' a conic is. e=0 is a perfect circle. As e increases toward 1, ellipses get more elongated. Parabolas have e=1, and hyperbolas have e>1.
Planets orbit stars in elliptical paths (Kepler's First Law). The star sits at one focus of the ellipse, not the center.
a is the semi-major axis (half the longest diameter), b is the semi-minor axis (half the shortest diameter). If a=b, it's a circle.
Satellite dishes, car headlights, projectile motion, suspension bridge cables, and parabolic microphones all use parabolic shapes to focus signals or reflect paths.
Lines the hyperbola approaches but never touches. For (x²/a²) - (y²/b²) = 1, the asymptotes are y = ±(b/a)x.
Yes! The general form Ax² + Bxy + Cy² + Dx + Ey + F = 0 represents all conics. The discriminant B² - 4AC determines which type.
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