Math notesCoordinateEllipse
Independent topic page
Ellipse full marks: standardise → identify a>b or major-axis orientation → e²=1−b²/a² → foci/directrices/tangent dictionary → verify 0<e<1.
01. If b>a, major axis is along y — rewrite before using (±ae,0) blindly.
02. Director circle x²+y²=a²+b² (locus of points with perpendicular tangents).
03. Parametric (a cosθ, b sinθ). Tangent at (x₁,y₁): xx₁/a²+yy₁/b²=1; slope form y=mx±√(a²m²+b²).
04. Never use hyperbola’s e²=1+b²/a² here — that single swap loses marks every year.
05. Advanced lane: orientation first — major axis along x or y decides (±ae,0) vs (0,±ae). Swapping this is a quiet, frequent full-mark leak.
06. Standardise the equation before the e,b²=a²(1−e²) dictionary; formula sheets on a non-standard form are noise.
Official · 2026
Official syllabus points for Ellipse only — not the full Coordinate portion map. Portion hub: Coordinate · Coordinate Geometry analysis.
No topic-specific exclusions flagged — finish the unit points below for Ellipse.
Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf
Exam focusStandard ellipse forms, eccentricity, and tangent/normal conditions.
Syllabus points
Do / check
2020–2026 paper weight
Topic match inside parent portion Coordinate Geometry (12.8% of Math tags). Stats below are portion backdrop; matched subtopics show where Ellipse sits.
Portion-wide type/difficulty mix lives on the Coordinate Geometry analysis — not repeated here so this page stays on Ellipse.
01. Dominant subtopics: Parabola (8), Circle (8), Straight line (5), Ellipse (5).
Full PCM trends board → · Coordinate Geometry subject page →
Cross-topic stems
Advanced rarely asks a chapter in isolation. These are tagged multi-idea Math stems that touch Ellipse — practise the knowledge bridge, not only the single-chapter drill.
Step 1: solve the two ellipse equations simultaneously for the first-quadrant intersection P. Step 2: differentiate implicitly to get slopes of both tangents at P; convert the angle formula into 4 tan θ. Step 3: for the common area, set up the integral (or standard ellipse-sector form) of the overlapping region and read cot α from that area.
Two different asks on one geometry figure — angle then area — under numerical fatigue.
For two conics, always compute intersection → slopes → area in that fixed order.
Step 1: recover points P, Q, R from the slope/tangent conditions on the parabola. Step 2: find the circumcircle of triangle PQR (perpendicular bisectors). Step 3: read the required radius / relation that also ties to an ellipse option.
Geometry hops across three curve types in one SC item.
Practise ‘points on parabola → circle through them → conic check’ as a single drill.
Step 1: for each List-I geometric condition, reduce to a standard conic equation or point. Step 2: match coordinates/parameters in List-II without mixing ellipse and hyperbola templates. Step 3: verify one substituted point per match to avoid swap errors.
Matching lists punish template mix-ups under time pressure.
Keep a one-page conics identity sheet and practise full matching sets weekly.
Step 1: place the ellipse in standard form and introduce eccentric angle or slope form. Step 2: impose the geometric condition (tangent/chord/focus). Step 3: match the simplified relation to the correct option.
Parametric slips create attractive wrong options.
Keep both slope and parametric forms of ellipse tangents fluent.
Step 1: for the ellipse–parabola common-tangent MC, write both tangent conditions and compare. Step 2: later area NUM — find intersection limits first. Step 3: integrate difference of functions; keep trig-equation NUM technique nearby for neighbouring items.
Classic Advanced join: geometry conditions feeding calculus numerics in the same paper.
Any week you revise conics, end with one area-between-curves NUM.
Why methods matter
Ellipse vs hyperbola confusion is a yearly mark killer. These methods lock e²=1− and major-axis orientation before foci — protect that habit.
Prevents (±ae,0) on a vertical major axis — a quiet, frequent error.
Do thisCompare a²,b² before writing foci.
Prevents (±ae,0) on a vertical ellipse.
x²/9+y²/25=1 → b=5>a=3 → foci on y-axis.
Blocks hyperbola’s 1+ formula from entering ellipse working.
Do thisSay ‘ellipse ⇒ 1 − (minor²)/(major²)’ aloud.
Blocks hyperbola formula intrusion.
x²/25+y²/16=1 → e²=1−16/25=9/25 → e=3/5, foci (±3,0).
Formula sheets only work after the equation is actually standard.
Do thisOnly after standard form, hit one entry.
Formula sheet fails on non-standard equations.
Divide through / complete shift first.
Tangency without contact point — and without mixing hyperbola’s condition.
Do thisUse c²=a²m²+b² condition for y=mx+c.
Tangency condition without finding point of contact.
Compare with hyperbola c²=a²m²−b² to avoid mix-ups.
Processed examples
Read the impact line first. Then notice which family/model was chosen. The solution is written the way your rough sheet should look in the hall.
For x²/25 + y²/16 = 1, find e and foci.
Lock e²=1− (not 1+) before writing foci — hyperbola contamination is common.
Model chosena=5,b=4 → e²=1−16/25.
Answere=3/5; foci (±3,0)
Takeaway: 0<e<1 sanity check after computing e.
Show that y=mx+√(a²m²+b²) touches the ellipse x²/a²+y²/b²=1.
Slope tangency condition is a dictionary hit — don’t re-derive every time.
Model chosenc² = a²m²+b² for y=mx+c.
AnswerTangency via c²=a²m²+b²
Takeaway: Ellipse uses +, hyperbola uses − under the root.
Find all lines through (5, 0) that are tangent to the ellipse x²/9 + y²/4 = 1.
External-point + ellipse tangency condition is a yearly parameterised-line pattern.
Model choseny = m(x − 5); impose c² = a²m² + b².
Answery = (x − 5)/2 and y = −(x − 5)/2
Takeaway: Combine fixed-point form with the ellipse c² = a²m² + b² condition.
Reusable models
Advanced stems cluster. Name the family in 10–15s, write the fast model, then run the same path. Do not invent a new plan each time.
Looks likeFind eccentricity or foci/directrices
Fast modelMajor axis first; e²=1−(minor²)/(major²); verify 0<e<1.
Orientation + 1− lock prevents the hyperbola-formula trap.
Do not use e²=1+b²/a² on an ellipse.
Looks likeLine touches ellipse; y=mx+c tangency
Fast modelc²=a²m²+b² (standard) or xx₁/a²+yy₁/b²=1 at point.
Condition without finding contact point.
Do not use hyperbola’s c²=a²m²−b².
Looks likePerpendicular tangents; geometric claims
Fast modelDirector circle x²+y²=a²+b²; each claim independent T/F.
Named properties kill options without computation.
Do not re-derive director circle in the hall unless asked.
Looks likeParametric (a cosθ, b sinθ); eccentric angle; corresponding point on aux circle
Fast modelUse eccentric angle θ; map to auxiliary circle x²+y²=a² when helpful.
θ-language shortens chord and tangent stems.
Do not force Cartesian when parametric is natural.
Hall timing
Write a², b² explicitly.
Confirm 0<e<1. Directrices x=±a/e for major along x.
For MC, test each geometric claim independently.
If e≥1 in your working, you used the wrong formula.
After the model is locked
x²/a²+y²/b²=1; e²=1−b²/a²; foci (±ae,0)
Major axis along x
x²/a²+y²/b²=1 with b>a; e²=1−a²/b²; foci (0,±be)
Vertical major axis
x=±a/e (major along x)
Directrix asks
(a cosθ, b sinθ)
Point on ellipse
xx₁/a² + yy₁/b² = 1
Tangent at (x₁,y₁)
y=mx±√(a²m²+b²)
Tangent with slope m
x²+y²=a²+b²
Perpendicular tangents locus
PF₁+PF₂=2a
Definition checks
Make it automatic