MathematicsChapter 7: Conic Section
Tangent To An Ellipse
At (x1,y1) on x^2/a^2+y^2/b^2=1, tangent is xx1/a^2+yy1/b^2=1.
Learning objectives
- State the rule for tangent to an ellipse.
- Solve a tangent to an ellipse problem step by step.
- Check the result, restrictions, notation, and units.
Definition
At (x1,y1) on xfrac2a2+yfrac2b2=1, tangent is xx1/a2+yy1/b2=1.
Where / symbols
- Tangent to an ellipse
- At (x1,y1) on x^2/a^2+y^2/b^2=1, tangent is xx1/a^2+yy1/b^2=1.
CONCEPT AND RULE
At (x1,y1) on xfrac2a2+yfrac2b2=1, tangent is xx1/a2+yy1/b2=1.
METHOD
For tangent to an ellipse, 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
At (3,4) on xfrac225+yfrac216=1 is 3x/25+y/4=1.
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 tangent to an ellipse
- 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:
- At (3,4) on xfrac225+yfrac216=1 is 3x/25+y/4=1.
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
- Tangent to an ellipse
- rule
- worked example
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Tangent To An Ellipse
Sticky Notes · Formula and Worked Revision
Core Idea
- At (x1,y1) on xfrac2a2+yfrac2b2=1, tangent is xx1/a2+yy1/b2=1.
Rule / Formula
- At (x1,y1) on xfrac2a2+yfrac2b2=1, tangent is xx1/a2+yy1/b2=1.
Method
- Identify givens and restrictions.
- Write the rule.
- Substitute and simplify.
- Check the result.
Worked Example
- At (3,4) on xfrac225+yfrac216=1 is 3x/25+y/4=1.
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
- At (x1,y1) on xfrac2a2+yfrac2b2=1, tangent is xx1/a2+yy1/b2=1.
- Worked pattern: At (3,4) on xfrac225+yfrac216=1 is 3x/25+y/4=1.
- Final check: domain, sign, notation, units, and reasonableness.
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