Parabolas and rectangular hyperbolas
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| parabola/pəˈræbələ/ | 抛物线 | pāo wù xiàn |
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 parabola 抛物线: The locus of points equally distant from a fixed focus and a fixed directrix.
Choose the mathematical structure
- For y²=4ax, the focus is (a,0), directrix x=-a and vertex (0,0). For xy=c², the coordinate axes are asymptotes. Read the parameter from the coefficient rather than assuming it equals 4a.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines parabola?
The locus of points equally distant from a fixed focus and a fixed directrix.
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 y²=12x, a=3: the focus is (3,0) and directrix x=-3. At x=3, y=±6; at (3,6), distance to the focus and directrix is 6. For xy=9, the point (3,3) lies on the hyperbola and x=0,y=0 are its asymptotes.
Parabolas and rectangular hyperbolas
For y²=4ax, the focus is (a,0), directrix x=-a and vertex (0,0)
Compare the model with the worked case and explain one change.
For y²=12x, find the focus x-coordinate.
Compare 4a=12, so a=3 and focus x-coordinate is 3.
Test a tempting shortcut
- The coefficient of x is 4a, not a. Include both signs when solving y². An asymptote is approached; it is not a finite intercept of xy=c².
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
For y²=4ax, the focus x-coordinate is 4a. This claim is false. Explain which definition or assumption it violates.
At x=3 on that parabola, find the positive y-coordinate.
At x=3, y²=36; choose the positive root y=6.
For y²=4ax, the focus x-coordinate is 4a.
The coefficient of x is 4a, not a. Include both signs when solving y². An asymptote is approached; it is not a finite intercept of xy=c².
Interpret a new situation
- FP1 coordinate-geometry support for parabolas and rectangular hyperbolas. Ellipse area and ellipse parametrisation are excluded.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
For xy=9 with x=3, find y.
From xy=9 and x=3, y=9/3=3.
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 pure mathematics; official unit FP1. 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.
The locus of points equally distant from a fixed focus and a fixed directrix. Choose the relationship, show the method, check its assumptions and interpret the result.