Conics, parametric curves and tangent reasoning
| English | 中文 | Pinyin |
|---|---|---|
| parametric equation/ˌpærəˈmetrɪk ɪˈkweɪʒn/ | 参数方程 | cān shù fāng chéng |
How can a rotating parameter trace an ellipse?
- An ellipse can be traced by a rotating parameter even when it is awkward to describe as a single y=f(x).
- This lesson studies parametric equation 参数方程: A representation in which coordinates are expressed using a common parameter.
Choose the mathematical structure
- For x=a cos t,y=b sin t, eliminating t gives x²/a²+y²/b²=1. For parametric curves, dy/dx=(dy/dt)/(dx/dt), where dx/dt≠0. A zero denominator may signal a vertical tangent.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines parametric equation?
A representation in which coordinates are expressed using a common parameter.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
For x=3cos t,y=2sin t at t=π/4, the gradient is (2cos t)/(-3sin t)=-2/3. The point is (3/√2,√2). The area inside the ellipse is πab=6π; its semiaxes are 3 and 2.
Find the gradient of x=3cos t,y=2sin t at t=π/4.
Divide the parameter derivatives: dy/dx=2 cos t/(-3 sin t)=-2/3 at π/4.
Test a tempting shortcut
- Eliminating a parameter may lose a domain restriction or direction of travel. A vertical tangent cannot be assigned a finite dy/dx. Distinguish semiaxes from full widths.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
Eliminating a parameter always preserves every restriction and direction automatically. This claim is false. Explain which definition or assumption it violates.
Find the ellipse area divided by π for a=3,b=2.
Ellipse area=πab=6π, so area/π=6.
Eliminating a parameter always preserves every restriction and direction automatically.
Eliminating a parameter may lose a domain restriction or direction of travel. A vertical tangent cannot be assigned a finite dy/dx. Distinguish semiaxes from full widths.
Interpret a new situation
- For advanced coordinate geometry, use the defining equation and a consistent parameter. For IAL P4 parametric integration, use the specification's restricted requirements rather than importing all Further Pure conics.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find the full horizontal width of that ellipse.
The horizontal semiaxis is 3, so full width is 6.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- edexcel IAL further mathematics; official unit FP3. Other-unit enrichment is identified in the scope review; it is not an extra cash-in requirement.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A representation in which coordinates are expressed using a common parameter. Choose the relationship, show the method, check its assumptions and interpret the result.