Conics, parametric curves and tangent reasoning · 圆锥曲线、参数曲线及切线推理
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| 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? · 哪个描述正确定义了参数方程？
A representation in which coordinates are expressed using a common parameter. · 坐标用公共参数表示的一种表达形式。
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. · 求 x=3cos t, y=2sin t 在 t=π/4 处的斜率。
Divide the parameter derivatives: dy/dx=2 cos t/(-3 sin t)=-2/3 at π/4. · 参数导数相除：dy/dx=2 cos t/(-3 sin t)=-2/3 在 π/4 处。
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. · 求椭圆面积除以 π 的值，其中 a=3, b=2。
Ellipse area=πab=6π, so area/π=6. · 椭圆面积=πab=6π，因此面积/π=6。
Eliminating a parameter always preserves every restriction and direction automatically. · 消参总是能自动保留所有限制条件和方向。
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. · 消参可能会丢失定义域限制或运动方向。垂直切线的 dy/dx 无法赋予有限值。注意区分半轴长与完整宽度。
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. · 求该椭圆的完整水平宽度。
The horizontal semiaxis is 3, so full width is 6. · 水平半轴长为 3，故完整宽度为 6。
Match each part of a complete solution to its purpose. · 将完整解答的每个部分与其目的相匹配。
An assumption justifies the model; a check tests the result; interpretation connects it to the question. · 假设用于论证模型的合理性；检查用于验证结果；解释用于将其与问题建立联系。
Use this in your course
- edexcel IAL pure mathematics; official unit P4. 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.