MathematicsChapter 7: Conic Section
Conics and Eccentricity
A conic is defined by a focus-directrix ratio e; ellipse has e<1, parabola e=1, hyperbola e>1.
Learning objectives
- State the rule for conics and eccentricity.
- Solve a conics and eccentricity problem step by step.
- Check the result, restrictions, notation, and units.
Definition
A conic is defined by a focus-directrix ratio e; ellipse has e<1, parabola e=1, hyperbola e>1.
Formula
PF=e times distance to the directrix classi
Where / symbols
- Conics and eccentricity
- A conic is defined by a focus-directrix ratio e; ellipse has e<1, parabola e=1, hyperbola e>1.
CONCEPT AND RULE
A conic is defined by a focus-directrix ratio e; ellipse has e<1, parabola e=1, hyperbola e>1.
METHOD
For conics and eccentricity, identify the given information and required result. Select the matching definition or formula, state its conditions, substitute carefully, simplify one line at a time, and check.
WORKED EXAMPLE
The locus PF=e times distance to the directrix classifies the conic.
COMMON MISTAKE
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
How to solve conics and eccentricity
- Identify the mathematical objects, givens, restrictions, and required result.
- Choose and write the relevant definition, theorem, or formula.
- Substitute with brackets and carry out one justified step per line.
- Simplify, state the final answer clearly, and verify it in the original setting.
Worked example
- Worked example:
- The locus PF=e times distance to the directrix classifies the conic.
In simple words
Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
Key points
- Conics and eccentricity
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Conics and Eccentricity
Sticky Notes · Formula and Worked Revision
Core Idea
- A conic is defined by a focus-directrix ratio e; ellipse has e<1, parabola e=1, hyperbola e>1.
Rule / Formula
- A conic is defined by a focus-directrix ratio e; ellipse has e<1, parabola e=1, hyperbola e>1.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- The locus PF=e times distance to the directrix classifies the conic.
Common Error
- Write the governing rule before substitution, keep brackets and signs visible, and check that the answer satisfies all domain or interval conditions.
FBISE Exam Tip
- Show the formula, substitutions, algebra, restrictions, and final answer; method marks need visible reasoning.
60-second revision
- A conic is defined by a focus-directrix ratio e; ellipse has e<1, parabola e=1, hyperbola e>1.
- Worked pattern: The locus PF=e times distance to the directrix classifies the conic.
- Final check: domain, sign, notation, units, and reasonableness.
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