MathematicsChapter 7: Conic Section
Conic Intersections
Substitute one curve into another and use real roots plus coordinate checks.
Learning objectives
- State the rule for conic intersections.
- Solve a conic intersections problem step by step.
- Check the result, restrictions, notation, and units.
Definition
Substitute one curve into another and use real roots plus coordinate checks.
Where / symbols
- Conic intersections
- Substitute one curve into another and use real roots plus coordinate checks.
CONCEPT AND RULE
Substitute one curve into another and use real roots plus coordinate checks.
METHOD
For conic intersections, 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
A line tangent to a conic produces a repeated root after substitution.
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 conic intersections
- 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:
- A line tangent to a conic produces a repeated root after substitution.
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
- Conic intersections
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Conic Intersections
Sticky Notes · Formula and Worked Revision
Core Idea
- Substitute one curve into another and use real roots plus coordinate checks.
Rule / Formula
- Substitute one curve into another and use real roots plus coordinate checks.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- A line tangent to a conic produces a repeated root after substitution.
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
- Substitute one curve into another and use real roots plus coordinate checks.
- Worked pattern: A line tangent to a conic produces a repeated root after substitution.
- Final check: domain, sign, notation, units, and reasonableness.
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