Math notesCoordinateHyperbola
Independent topic page
Hyperbola full marks: standardise → e²=1+b²/a² → foci → asymptotes y=±(b/a)x → tangent dictionary — never borrow ellipse’s e formula.
01. e>1 always for hyperbola. If your e comes out <1, you swapped formulae.
02. Asymptotes are the fastest recognition/check tool in MC.
03. Tangent: xx₁/a²−yy₁/b²=1; slope form y=mx±√(a²m²−b²) (require a²m²−b²>0).
04. Conjugate hyperbola and rectangular (a=b) cases: keep asymptotes perpendicular for rectangular.
05. Advanced lane: e lock is 1+ — if your eccentricity comes out less than 1, you used the ellipse page by mistake. Separate revision stops interference.
06. Asymptotes often appear as MC options; write them as soon as the standard form is clear. Slope domain for tangents deletes impossible real m early.
Official · 2026
Official syllabus points for Hyperbola only — not the full Coordinate portion map. Portion hub: Coordinate · Coordinate Geometry analysis.
No topic-specific exclusions flagged — finish the unit points below for Hyperbola.
Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf
Exam focusStandard hyperbola forms, asymptotes, and eccentricity 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 Hyperbola sits.
Portion-wide type/difficulty mix lives on the Coordinate Geometry analysis — not repeated here so this page stays on Hyperbola.
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 Hyperbola — practise the knowledge bridge, not only the single-chapter 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.
Why methods matter
Hyperbola needs e²=1+ and asymptotes early. These methods keep ellipse formulae out of your working — that separation is the whole game.
If e comes out <1, you used the ellipse formula — this lock prevents that.
Do thisHyperbola ⇒ e²=1+b²/a² spoken before computing.
Stops ellipse formula contamination.
If e comes <1, restart with 1+ not 1−.
Many MC options are asymptote equations; writing them early wins time.
Do thisWrite y=±(b/a)x immediately after standard form.
Many MC options are asymptote equations.
Even when not asked, asymptotes help sanity-check orientation.
Deletes impossible real tangents before you commit algebra.
Do thisFor y=mx±√(a²m²−b²), require a²m²≥b².
Prevents imaginary tangents presented as real options.
Compare mentally with ellipse √(a²m²+b²) always real.
Interference errors drop when hyperbola is revised as its own language.
Do thisRevise hyperbola on its own day; do not alternate formulae in one sitting without a checklist.
Interference errors drop sharply.
Keep a two-column flash: ellipse 1− vs hyperbola 1+.
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 x²/9 − y²/16 = 1, find e and asymptotes.
Asymptotes should appear on rough sheet immediately after standard form.
Model chosene²=1+16/9; asymptotes y=±(4/3)x.
Answere=5/3; y=±(4/3)x
Takeaway: e>1 check; write asymptotes early.
For which m does y=mx+√(a²m²−b²) make sense as a real tangent to x²/a²−y²/b²=1?
Domain a²m²≥b² deletes impossible options before algebra.
Model chosenNeed a²m²−b² ≥ 0.
Answer|m| ≥ b/a
Takeaway: Check slope domain before writing tangents.
Find the slopes of all tangents from (0, 3) to the hyperbola x²/9 − y²/4 = 1.
Hyperbola uses c² = a²m² − b² — ellipse’s + under the root is a designed trap.
Model choseny = mx + 3; impose c² = 9m² − 4.
Answerm = ±√13 / 3
Takeaway: Keep hyperbola’s minus-sign tangency condition separate from ellipse.
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 likeEccentricity, foci, asymptotes
Fast modele²=1+b²/a²; write asymptotes y=±(b/a)x immediately; verify e>1.
Asymptotes appear constantly in options — write them early.
Do not use ellipse’s 1− formula.
Looks likeTangents of slope m; condition on c
Fast modely=mx±√(a²m²−b²) only when a²m²≥b².
Domain check eliminates impossible options instantly.
Do not use √(a²m²+b²) (ellipse) on hyperbola.
Looks likeRectangular hyperbola; asymptotes perpendicular; xy=c² joins
Fast modela=b ⇒ e=√2; asymptotes ⊥ — use as instant T/F.
Two facts decide many MC stems.
Do not treat every hyperbola as rectangular.
Looks likeConjugate hyperbola; e and e′ with 1/e²+1/e′²=1; asymptote-based MC
Fast modelWrite conjugate form and asymptotes first; then apply the named relation.
Named relations beat re-deriving from scratch.
Do not mix conjugate formulae with ellipse director-circle habits.
Hall timing
x²/a²−y²/b²=1 vs y²/a²−x²/b²=1 changes foci axis.
Asymptotes for standard: y=±(b/a)x.
Check domain of slope form (a²m²>b²).
Independent T/F for MC property lists.
After the model is locked
x²/a²−y²/b²=1; e²=1+b²/a²; foci (±ae,0)
Transverse along x
y=±(b/a)x
Standard hyperbola
x=±a/e
Directrix asks
(a secθ, b tanθ) or (a cosh t, b sinh t) as used
Point on hyperbola
xx₁/a² − yy₁/b² = 1
Tangent at (x₁,y₁)
y=mx±√(a²m²−b²)
Tangent with slope m (when defined)
|PF₁−PF₂|=2a
Definition checks
a=b; asymptotes perpendicular; e=√2
Rectangular hyperbola
Make it automatic