Math101 · Independent Ontario learning library
Math101learn.math101.caConic Sections
Conic sections are curves obtained by intersecting a double cone with a plane: circles, ellipses, parabolas, and hyperbolas.
Conics model planetary orbits, projectiles, reflectors, navigation, and quadratic equations. Their multiple definitions connect geometry, algebra, and physical applications.
Intuition and core definition
Conic sections are curves obtained by intersecting a double cone with a plane: circles, ellipses, parabolas, and hyperbolas. They can also be defined as loci using distances to foci and directrices. The plane’s angle and position determine which curve appears.
Notation, language, and conditions
In general quadratic form $Ax^2+Bxy+Cy^2+Dx+Ey+F=0$, the discriminant $B^2-4AC$ helps classify nondegenerate conics after accounting for rotation: negative suggests ellipse/circle, zero parabola, positive hyperbola. Standard forms expose centres, vertices, axes, and parameters more directly.
Why this idea matters
Conic classification connects the geometry of slicing a cone with algebraic patterns in quadratic equations.
A dependable method
- Identify whether the information is geometric, locus-based, or algebraic.
- For an equation, group quadratic and linear terms and inspect signs and coefficients.
- Complete squares and rotate axes if an $xy$ term requires it at the studied level.
- Match to a standard form and read defining features.
- Check the equation with vertices, symmetry, and domain behaviour.
Worked example
Representations and interpretation
A physical cone-plane model shows the family: a perpendicular cut gives a circle, an oblique closed cut an ellipse, a parallel-to-generator cut a parabola, and a cut through both nappes a hyperbola. Coordinate graphs encode the same curve families.
Reasoning about variations
Degenerate equations can produce a point, pair of lines, one line, or no real locus, so sign inspection alone is not a complete classification. Standard-form conversion exposes these cases.
Common mistakes
How to check your work
- Substitute standard-form vertices into the equation.
- Inspect symmetry under $x\mapsto-x$ or $y\mapsto-y$ when centred at the origin.
- Use the quadratic discriminant as a classification cross-check, not the only evidence.
Practice
- Classify $x^2/16-y^2/9=1$.
- Classify $y^2=8x$.
- When is an origin-centred ellipse a circle?
Answers and brief solutions
Show answers
- Hyperbola The squared terms have opposite signs.
- Parabola Only one variable is squared.
- When its two semi-axis lengths are equal Equal denominators produce equal radius in every direction.
Synthesis and transfer
A satellite dish, planetary orbit, and cooling-tower profile use different conics; matching each physical constraint to focus or symmetry properties explains why one family fits better than another.
A satellite orbit is modelled by an ellipse with the attracting body at a focus, not at an arbitrary centre point. A headlight reflector uses a parabola because rays from its focus leave parallel to the axis, while time-difference location data naturally produce hyperbolas. These applications depend on focus or directrix properties, not merely on the visual resemblance of a curve. Algebraic classification supplies another lens: equal-sign squared terms suggest a closed ellipse-type curve, one squared variable a parabola, and opposite signs a hyperbola, subject to degeneracy and rotation. Connecting the physical constraint to the defining locus avoids choosing a conic by shape alone.
Related topics
Teaching and accessibility note
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Classify $x^2/16-y^2/9=1$.
- The squared terms have opposite signs.
End of lesson
Nice work making it this far.
Understanding grows through return visits. Save this lesson, try the practice, or continue when you are ready.
