Math notesCoordinateParabola
Independent topic page
Parabola full marks: standardise to y²=4ax (or x²=4ay) → write a, focus, directrix → parametric/tangent tool → eliminate t for locus.
01. y²=4ax: focus (a,0), directrix x=−a, latus rectum length 4a, parametric (at²,2at).
02. x²=4ay: focus (0,a), directrix y=−a, parametric (2at,at²).
03. Tangent at (x₁,y₁): yy₁=2a(x+x₁) for y²=4ax; slope form y=mx+a/m.
04. Focal chord: ends t₁,t₂ satisfy t₁t₂=−1. Mid-point locus problems are eliminate-t drills.
05. Advanced lane: extract a (and vertex shift) before focus/directrix/tangent dictionary. Wrong a breaks every focal property at once.
06. Moving-point / chord stems: parametric (at²,2at) keeps degree low; Cartesian often explodes mid-paper.
Official · 2026
Official syllabus points for Parabola only — not the full Coordinate portion map. Portion hub: Coordinate · Coordinate Geometry analysis.
No topic-specific exclusions flagged — finish the unit points below for Parabola.
Source: https://jeeadv.ac.in/documents/jee-advanced-2026-syllabus.pdf
Exam focusStandard parabola forms, focal properties, and tangent 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 Parabola sits.
Portion-wide type/difficulty mix lives on the Coordinate Geometry analysis — not repeated here so this page stays on Parabola.
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 Parabola — practise the knowledge bridge, not only the single-chapter drill.
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: write the parabola in standard form and express the geometric condition algebraically. Step 2: reduce to an equation in the unknown parameter. Step 3: extract the required integer/decimal for the NUM grid.
Conic geometry collapses into algebra; diagram alone is not enough.
Translate every parabola condition into an equation before computing.
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.
Step 1: for the area item, find intersection limits and integrate. Step 2: for the parabola item, apply focus–directrix or tangent properties. Step 3: notice how often Advanced seats Calculus and Conics as neighbours — train that adjacency.
2020’s Calculus weight makes area items feel easy until limits are wrong.
Always find limits from geometry before writing ∫.
Why methods matter
Parabola fails when ‘a’ is wrong. These methods extract a from 4a first and prefer parametric for moving points — that is the scoring pattern.
Wrong a breaks focus, directrix, and tangents together — extract before dictionary.
Do thisFrom y²=8x get 4a=8 ⇒ a=2 before any formula.
Prevents systematic wrong focus/tangent.
Focus becomes (2,0), not (8,0).
Moving-point stems stay low-degree instead of exploding in Cartesian.
Do thisUse t-parameters for chords/locus; eliminate at end.
Avoids degree explosion in Cartesian.
Mid-point (h,k) in terms of t₁,t₂ → use relations → locus in h,k.
Stops re-deriving PF=PM every time — exam speed comes from recall after standard form.
Do thisAfter standard form, recall only the entry needed.
Saves re-deriving PF=PM each time.
Ask focus? Write (a,0). Ask tangent slope 1? y=x+a.
Translated parabolas stop tricking you into unshifted focus coordinates.
Do thisComplete shift to vertex origin before standard dictionary.
Translated parabolas trick students who reuse unshifted focus.
Replace X=x−h, Y=y−k, apply standard, translate back.
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 y²=8x, find focus and the tangent with slope 1.
Extract a from 4a — wrong a breaks every dictionary entry.
Model chosen4a=8 ⇒ a=2; then focus and y=mx+a/m.
AnswerFocus (2,0); tangent y=x+2
Takeaway: a from 4a, always.
For y²=4ax, show that if chord joins t₁ and t₂ and is focal, then t₁t₂=−1.
Focal chord condition is a reusable model for many locus stems.
Model chosenChord through focus (a,0) ⇒ substitute into chord equation.
Answert₁t₂ = −1
Takeaway: Write t₁t₂=−1 early on focal-chord stems.
A focal chord of y² = 4ax has one endpoint with ordinate 4a. Find the other endpoint in terms of a.
t₁t₂ = −1 turns a wordy focal-chord stem into one arithmetic step.
Model chosenParametric (at², 2at); t₁ from ordinate; t₂ = −1/t₁.
Answer(a/4, −a)
Takeaway: Write t₁t₂ = −1 immediately on every focal-chord stem.
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 likeGiven parabola equation; focus, directrix, tangent
Fast modelExtract a from 4a; write focus/directrix; hit tangent form.
Wrong a is the main failure — extract first.
Do not use focus (a,0) with a equal to the coefficient of x.
Looks likeChord joining t₁,t₂; focal chord; mid-point
Fast modelUse (at²,2at); focal ⇒ t₁t₂=−1; eliminate parameters for locus.
Parametric beats heavy Cartesian for moving points.
Do not expand chord equations in x,y without parameters when t is natural.
Looks likeTangent with slope m; pair of tangents
Fast modelFor y²=4ax, write y=mx+a/m directly.
Dictionary hit — no point-of-contact hunt unless asked.
Do not derive tangent from first principles each time.
Looks likeLocus of mid-point of focal chord; moving tangent condition
Fast modelWrite parametric relation → eliminate t → recognise conic/line.
Elimination template is reusable across conics.
Do not expand fully in x,y before writing the parameter condition.
Hall timing
Shifted vertex? Translate first. Wrong a is fatal.
Moving points → parametric. Fixed slope → y=mx+a/m.
Focal chord: use t₁t₂=−1 early.
Plug a known parametric point onto tangent if unsure.
After the model is locked
Focus (a,0); directrix x=−a; parametric (at²,2at)
Opens right
Focus (0,a); directrix y=−a; parametric (2at,at²)
Opens up
yy₁=2a(x+x₁) for y²=4ax
Tangent at (x₁,y₁)
y=mx+a/m
Tangent with slope m
ty=x+at²
Parametric tangent
t₁t₂=−1
Chord through focus
Length 4a
LR asks
Make it automatic