3.4.2
Intuition Har parabola ke peeche ek hi ideaParabola un sabhi points ka set hai jo ek fixed point (focus) aur ek fixed line (directrix) se equidistant hote hain .
YE kyun zaroori hai: neeche diya hua har formula bas is ek sentence ko coordinates mein likha hua hai. Agar tumhe sirf "distance to focus = distance to directrix" yaad hai, toh tum scratch se charon standard forms rebuild kar sakte ho — kuch bhi memorize karne ki zaroorat nahi.
Ek fixed point F focus ) aur ek fixed line ℓ directrix ) diya hua hai jo F P distance ( P , F ) = distance ( P , ℓ ) . e = 1
PARTS kya hain?
Focus F Directrix — woh special line.Axis — focus se guzarne wali line, directrix ke perpendicular (symmetry ki axis).Vertex — axis pe parabola ka woh point jo focus aur directrix ke beech exactly midway hota hai.Latus rectum — focus se guzarne wala chord, directrix ke parallel.
KAISE — convenient coordinates set karo. Vertex ko origin pe rakho, axis x F = ( a , 0 ) a > 0 x = − a
Maano P = ( x , y )
Focus tak distance: P F = ( x − a ) 2 + y 2
Directrix x = − a = ∣ x + a ∣
Inhe equal karo (definition), phir square karo:
( x − a ) 2 + y 2 = ( x + a ) 2
Square kyun karte hain? Square root aur absolute value dono ko cleanly khatam karne ke liye — dono sides non-negative hain.
Expand karo:
x 2 − 2 a x + a 2 + y 2 = x 2 + 2 a x + a 2
x 2 a 2 y 2 = 4 a x
YE kyun jaanna zaroori hai? Ye batata hai ki parabola focus pe kitna "wide" hai — sketch banane mein kaam aata hai.
Latus rectum woh chord hai jo F = ( a , 0 ) x = a y 2 = 4 a x y 2 = 4 a ⋅ a = 4 a 2 ⟹ y = ± 2 a .
Toh endpoints hain ( a , 2 a ) ( a , − 2 a ) LR = 2 a − ( − 2 a ) = 4 a .
Sign badalne aur usi derivation mein x ↔ y Hamesha a > 0 ; equation mein sign direction control karta hai.)
Equation
Opens
Focus
Directrix
Axis
LR
y 2 = 4 a x Right
( a , 0 ) x = − a y = 0 4 a
y 2 = − 4 a x Left
( − a , 0 ) x = a y = 0 4 a
x 2 = 4 a y Up
( 0 , a ) y = − a x = 0 4 a
x 2 = − 4 a y Down
( 0 , − a ) y = a x = 0 4 a
Intuition Equation ko map ki tarah padhna
Agar y 2 x
Agar x 2 y
Focus usi side pe hota hai jis taraf parabola khulta hai. Sign + ⇒ − ⇒
y 2 = 12 x Step 1: y 2 = 4 a x 4 a = 12 ⇒ a = 3 Kyun? x hi 4 a a Step 2: Right ki taraf khulta hai (y 2 = ( a , 0 ) = ( 3 , 0 ) Step 3: Directrix x = − a ⇒ x = − 3 Step 4: Axis y = 0 = 4 a = 12
Worked example Ex 2 — Parabolax 2 = − 8 y Step 1: x 2 = − 4 a y 4 a = 8 ⇒ a = 2 Negative kyun? x 2 down ki taraf khulta hai.
Step 2: Focus = ( 0 , − a ) = ( 0 , − 2 ) y = a = 2 Step 3: Axis x = 0 = 8
Worked example Ex 3 — Data se equation banaoFocus ( 0 , 4 ) y = − 4 Step 1: Focus origin ke upar, directrix neeche → up ki taraf khulta hai, form x 2 = 4 a y Kyun? Vertex midway pe hai → ( 0 , 0 ) ( 0 , a ) a = 4 Step 2: x 2 = 4 ( 4 ) y ⇒ x 2 = 16 y . Check: directrix honi chahiye y = − a = − 4
Worked example Ex 4 — Ek point + definition se reverse karnaFocus ( 2 , 0 ) x = − 2 Step: ( x − 2 ) 2 + y 2 = ∣ x + 2∣ ( x − 2 ) 2 + y 2 = ( x + 2 ) 2 ⇒ y 2 = 8 x Ye kyun kaam karta hai: humne standard form assume nahi ki — humne ise derive kiya, jo confirm karta hai a = 2 = 8
y 2 = 12 x a = 12 Ye sahi kyun lagta hai: a Fix: coefficient 4 a a 4 a = 12 ⇒ a = 3
y 2 = 4 a x x = a Ye sahi kyun lagta hai: tum a ( a , 0 ) Fix: directrix vertex ke focus wali side se opposite side pe hoti hai, isliye x = − a
x 2 = 4 a y x Ye sahi kyun lagta hai: tum "squared variable = direction" match karte ho.
Fix: ye ulta hai — non-squared variable ki axis hi axis of symmetry hai. x 2 up/down khulta hai (axis = y
a < 0 Ye sahi kyun lagta hai: algebra mein a Fix: convention mein a > 0 equation mein sign direction set karta hai. a
Recall Quick self-test (answers chhupao, pehle forecast karo!)
Parabola ko define karne wali ek property kya hai? → focus aur directrix se equidistant.
Vertex focus aur directrix ke relative kahan hota hai? → midpoint pe.
y 2 = 4 a x ( a , 0 ) x = − a Latus rectum ki length? → 4 a
x 2 = − 4 a y
Recall Feynman: ek 12-saal ke bachche ko samjhaoSocho ek point (the focus ) aur zameen pe ek seedhi fence (the directrix ) hai. Tum chalo is tarah ki tum hamesha point se utni hi door ho jitnी fence se . Jo curved path tum banate ho woh parabola hai — jaise satellite dish ya pheenki hui ball ka path. Point jitna fence ke paas hoga, curve utna tight hoga; jitna door hoga, utna wide failega. "Vertex" wahan hai jahan tum dono ke sabse paas ho — dono ke beech mein bilkul right.
Mnemonic Layout yaad rakhne ka tarika"FoVeD on the axis" — Fo cus, Ve rtex, D irectrix sab axis pe line up hote hain, is order mein (vertex beech mein). Aur "4a is the width" — 4 a hi latus rectum length hai.
Parabola ki defining property Har point focus aur directrix se equidistant hota hai (e = 1
Standard form opening right y 2 = 4 a x a > 0
y 2 = 4 a x Focus ( a , 0 ) x = − a
y 2 = 4 a x 4 a
y 2 = 4 a x ( a , 2 a ) ( a , − 2 a )
y 2 = 12 x a 4 a = 12 ⇒ a = 3
Agar equation mein x 2 y
x 2 = − 4 a y Neeche (downward) khulta hai; focus ( 0 , − a ) y = a
Vertex kahan hota hai Focus aur directrix ke beech midpoint pe, axis pe.
Focus ( 0 , 4 ) y = − 4 x 2 = 16 y
through F, parallel to directrix
Parabola: dist to F = dist to line
Axis: through F, perp to directrix
Vertex: midway F and directrix
Latus rectum: chord through F
