Conics, parametric curves and tangent reasoning
مُقدَّم| English | المعنى |
|---|---|
| parametric equation/ˌpærəˈmetrɪk ɪˈkweɪʒn/ | معادلة معلمية |
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.
The method
- 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.
For x=3cos t, y=2sin t, how is dy/dx found at t=π/4?
الصعوبة 1/5 · 100 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
The chain rule gives dy/dx=(dy/dt)/(dx/dt). At t=π/4 both cos and sin equal 1/√2, so the gradient is -2/3.
A worked case
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. Give it to 3 significant figures.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
Divide the parameter derivatives: dy/dx=2 cos t/(-3 sin t)=-2/3 at π/4.
Where it goes wrong
- 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.
Not true: Eliminating a parameter preserves every restriction and direction automatically. Elimination can lose restrictions: x=t², y=t gives x=y², but x≥0 and the direction of motion must be stated separately.
Find the ellipse area divided by π for a=3,b=2.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
Ellipse area=πab=6π, so area/π=6.
Eliminating a parameter preserves every restriction and direction automatically.
الصعوبة 1/5 · 100 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
Elimination can lose restrictions: x=t², y=t gives x=y², but x≥0 and the direction of motion must be stated separately.
Applying it
- 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.
Find the full horizontal width of that ellipse.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
The horizontal semiaxis is 3, so full width is 6.
Match each ellipse quantity for x²/9+y²/4=1 to its value.
الصعوبة 2/5 · 140 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
The denominators give the SQUARED semiaxes, and the ellipse area is π times their product.
In your course
- Pearson Edexcel International A Level Further Mathematics, unit FP3.
- Every paper allows a calculator, but method marks still need the expression or equation written before its value.
- Parametric equation: A representation in which coordinates are expressed using a common parameter.
- For x=a cos t,y=b sin t, eliminating t gives x²/a²+y²/b²=1.
- Watch for: eliminating a parameter may lose a domain restriction or direction of travel.
Write from an ellipse area of 12 pi
- An ellipse has a = 4 and b = 3.
- Find the area divided by pi.
An ellipse has semi-axes 4 and 3. What is its area divided by pi?
4 times 3 is 12.
What is the area divided by pi?
الصعوبة 1/5 · 100 عملة + مكافأة صغيرة عند الإكمال الأول اليوم
The product of the semi-axes is 12. Other options — 6 (the lesson ellipse); 8 (the full width).