Reflective property of parabola (application in telescopes, antennas)

3.4.3 · Maths › Conic Sections

YEH hai kya — reflective property?

WHY it matters: parabola ki shape ka curved mirror weak parallel signals (door ke tare ki roshni, satellite waves) ko collect karta hai aur unki saari energy ek tiny spot pe concentrate karta hai — focus pe — jahan aap apna detector ya antenna rakhte ho. Isko ulta chalao aur tumhare paas searchlight aa jaata hai.

HOW set up karte hain

Standard parabola lo $y^{2} = 4 a x, focus F = (a, 0), directrix x = - a .$

Hum focus–directrix definition aur law of reflection (angle of incidence = angle of reflection about the tangent) se yeh property scratch se derive karenge.

Step 1 — Defining property (WHY yeh hamara tool hai)

Parabola ki definition se, uske kisi bhi point $P$ ke liye: $P F = P M$ jahan $M$ , $P$ se directrix par dala gaya perpendicular ka foot hai. Yeh step kyon? Saara magic isi equal-distance fact se aata hai — ek incoming parallel ray horizontally directrix ki taraf chalti hai, isliye uski geometry $P M$ se tied hai.

Step 2 — Tangent ki slope (derive karo)

$y^{2} = 4 a x$ ko implicitly differentiate karo: $2 y \frac{d y}{d x} = 4 a ⟹ \frac{d y}{d x} = \frac{2 a}{y} .$ Yeh step kyon? Reflection tangent line ke baare mein hoti hai, isliye hume $P = (x_{1}, y_{1})$ pe tangent ki slope chahiye: $m_{t} = \frac{2 a}{y _{1}}$ .

Step 3 — $P$ pe do rays

Incoming ray: axis ( $x$ -axis) ke parallel hai, toh iska direction horizontal hai, slope $= 0$ .
Focal radius: $P = (x_{1}, y_{1})$ se $F = (a, 0)$ tak line $P F$ , slope $m_{P F} = \frac{y _{1} - 0}{x _{1} - a} .$

Step 4 — Prove karo ki tangent angle bisect karti hai

Maano $α$ = horizontal incoming ray aur tangent ke beech angle, aur $β$ = tangent aur focal radius $P F$ ke beech angle. Use karo $tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}} .$

Angle $α$ (horizontal, slope 0, tangent slope $\frac{2 a}{y _{1}}$ ke saath): $tan α = \frac{\frac{2 a}{y _{1}} - 0}{1 + 0} = \frac{2 a}{y _{1}} .$

Angle $β$ (tangent slope $\frac{2 a}{y _{1}}$ vs $m_{P F} = \frac{y _{1}}{x _{1} - a}$ ). $y_{1}^{2} = 4 a x_{1}$ use karo:

= \left|\frac{2a(x_1-a) - y_1^2}{y_1(x_1-a) + 2a y_1}\right|.$$ Numerator mein $y_1^2 = 4ax_1$ substitute karo: $$2a(x_1-a) - 4ax_1 = 2ax_1 - 2a^2 - 4ax_1 = -2a^2 - 2ax_1 = -2a(x_1+a).$$ Denominator: $y_1(x_1 - a + 2a) = y_1(x_1+a).$ $$\tan\beta = \left|\frac{-2a(x_1+a)}{y_1(x_1+a)}\right| = \frac{2a}{y_1}.$$ > [!formula] Result > $$\tan\alpha = \tan\beta = \frac{2a}{y_1} \implies \boxed{\alpha = \beta}$$ > Tangent, horizontal incoming ray aur focal radius ke saath **equal angles** banata hai. Law of reflection se, axis ke parallel aane wali ray reflect hokar **seedha focus se guzarti hai**. ∎ *"$x_1+a$" kyon cancel hota hai:* yeh upar aur neeche dono mein same factor hai — ek signal ki $F$ aur directrix bilkul balance mein hain, exactly wahi jo focus–directrix definition guarantee karta hai. ![[3.4.03-Reflective-property-of-parabola-(application-in-telescopes,-antennas).png]] --- ## Ek elegant alternative view (Feynman-style shortcut) > [!intuition] "String" argument > Focus $F$ ko tangent line ke across reflect karo — tum bilkul directrix pe, point $M$ pe land karte ho (kyunki $PF = PM$ aur $\alpha=\beta$). Toh tangent, $FM$ ka **perpendicular bisector** hai. Ek horizontal ray $M$ ki taraf jaati hai isliye reflect hokar $F$ ki taraf jaati hai. Yeh disguise mein *shortest-path* / Fermat principle hai: light equal-angle path leta hai. --- ## Worked Examples > [!example] Example 1 — Dhundo kahan parallel ray focus hoti hai > Ek parabolic mirror $y^2 = 12x$ hai. Ek horizontal ray $P=(3,6)$ pe hit karti hai. Reflected ray kahan jaati hai? > > **Step 1.** $y^2=12x$ ko $y^2=4ax$ se compare karo $\Rightarrow 4a=12 \Rightarrow a=3$. *Kyon?* Hume focus chahiye. > **Step 2.** Focus $F=(a,0)=(3,0)$. *Kyon?* Standard parabola ka focus $(a,0)$ hota hai. > **Step 3.** Check karo $P$ curve pe hai: $6^2=36=12\cdot3$ ✓. > **Step 4.** Reflective property se, reflected ray $F=(3,0)$ se guzarti hai. **Done.** > *Sanity:* yahan $P=(3,6)$ aur $F=(3,0)$ vertically aligned hain — yeh ray seedha neeche focus mein reflect hoti hai. > [!example] Example 2 — Numerically equal angles verify karo > Same parabola, point $P=(3,6)$, $a=3$. > - Tangent slope: $m_t = \frac{2a}{y_1}=\frac{6}{6}=1$. > - Incoming horizontal slope $=0$: $\tan\alpha = |(1-0)/(1)| = 1 \Rightarrow \alpha=45°$. > - Focal radius $PF$: $m_{PF} = \frac{6-0}{3-3}$ = **vertical** (undefined). Ek vertical line ka slope-1 tangent ke saath angle $= 45°$ hota hai. Toh $\beta = 45°$. ✓ $\alpha=\beta$. > *Yeh kyon matter karta hai:* concrete numbers algebra confirm karte hain — hamesha pehle forecast karo phir verify karo. > [!example] Example 3 — Satellite dish design karo > Ek dish ka cross-section $y^2 = 4ax$ hai aur $0.25$ m ki depth pe $2$ m wide hai. Receiver kahan rakhoge? > > **Step 1.** "2 m wide" ⇒ half-width $y=1$; depth ⇒ $x=0.25$. *Kyon?* Rim point $(0.25,\,1)$ hai. > **Step 2.** Plug in: $1^2 = 4a(0.25) = a \Rightarrow a=1$. > **Step 3.** Focus $(a,0)=(1,0)$ pe hai: receiver ko **axis ke saath vertex se 1 m door** rakho. > *Kyon:* saari parallel satellite waves wahan converge hoti hain — reflective property literally engineering spec hai. --- ## Applications (tech ke peeche WHY) | Device | Source/Detector at focus | Direction of travel | |---|---|---| | Reflecting telescope | Detector/eyepiece focus pe | Star light IN → focus | | Satellite dish / radio antenna | Receiver ("LNB") focus pe | Signal IN → focus | | Car headlight / torch | Bulb focus pe | Light OUT → parallel beam | | Solar cooker | Pot focus pe | Sunlight IN → concentrated heat | --- > [!mistake] Common errors ko steel-man karna > **Mistake A: "Focus se nikli rays wapas focus pe converge hoti hain."** > *Kyon sahi lagta hai:* symmetry — ellipses mein ek focus se rays dusre focus mein JAATI hain. *Fix:* parabola mein **sirf ek focus** hota hai (doosra "infinity pe" hota hai). Focus rays bahar parallel jaati hain; sirf *parallel-in* rays focus pe converge hoti hain. > > **Mistake B: "Ray normal ke baare mein reflect hoti hai, isliye main normal ke saath angles equate karta hoon — alag result aata hai."** > *Kyon sahi lagta hai:* physics kehta hai angle of incidence = reflection about the normal. *Fix:* yeh same baat hai! Tangent ke saath equal angles ⇔ normal ke saath equal angles (normal ⟂ tangent). Humne proof mein tangent use kiya kyunki uski slope $\frac{2a}{y}$ compute karna easy hai. > > **Mistake C: Focus $(4a,0)$ use karna ya $4a$ bhool jaana.** > *Kyon sahi lagta hai:* $4a$ (latus rectum length) aur focal distance ko mix up kar lena. *Fix:* $y^2=4ax$ mein, focus $(a,0)$ hai; $4a$ se match karke $a$ nikalo. --- > [!recall]- Feynman: 12-saal ke bacche ko samjhao > Imagine karo ek bunch of runners sab ek hi direction mein ek curved wall ki taraf jog kar rahe hain jo smile ki shape mein hai. Woh wall ek special "smile shape" (parabola) hai. Chahe koi runner wall pe kahaan bhi hit kare, sab bounce hokar **ek exact spot** pe daud ke jaate hain — jaise sab ek hi darwaze pe milte ho. Isliye satellite dish space se faint TV signals pakad sakta hai: poori dish unhe ek tiny box mein funnel karti hai. Isko flip karo: us spot pe bulb lagao aur saari roshni ek straight beam mein nikal jaati hai — yahi torch hai! > [!mnemonic] Yaad rakho > **"Parallel IN → Focus. Focus OUT → Parallel."** > Isko **"PIF–FOP"** bolo — *Parallel-In-Focus, Focus-Out-Parallel.* Tangent fair referee hai: woh dono rays ko **equal angles** deta hai. --- ## Active Recall > [!recall] Quick self-test > 1. Reflective property ek sentence mein batao. > 2. Proof mein $y^2=4ax$ differentiate kyon karte hain? > 3. $y^2 = 20x$ ka focus kahan hai? > 4. Torch mein bulb kahan rakha jaata hai aur kyon? #flashcards/maths Parabola ki reflective property kehti hai ::: axis ke parallel ray focus se reflect hoti hai (aur tangent incoming ray aur focal radius ke saath equal angles banata hai) $y^2=4ax$ ke liye, focus hota hai ::: $(a,0)$ pe $y^2=4ax$ ka $(x_1,y_1)$ pe tangent slope ::: $\dfrac{2a}{y_1}$ ($2y\,y'=4a$ se) Proof mein use ki gayi defining focus–directrix fact ::: $PF = PM$ (focus se doori = directrix se doori) Kyon normal ki bajay tangent ke baare mein reflect karte hain ::: tangent ke saath equal angles ⇔ normal ke saath equal angles, aur tangent slope compute karna easier hai Satellite dish mein receiver rakha jaata hai ::: focus pe (saari parallel signals wahan converge hoti hain) Car headlight mein bulb rakha jaata hai ::: focus pe, taaki emitted rays axis ke parallel nikal sakein $y^2=20x$ ka focus ::: $4a=20\Rightarrow a=5$, focus $(5,0)$ Angle proof mein factor $(x_1+a)$ kyon cancel hota hai ::: yeh numerator aur denominator dono mein identically appear karta hai, focus–directrix balance reflect karta hai Parabolic reflection ka mnemonic ::: PIF–FOP: Parallel-In→Focus, Focus-Out→Parallel --- ## Connections - [[Parabola - standard equation y^2=4ax]] - [[Focus and Directrix of a Conic]] - [[Tangent and Normal to a Conic]] - [[Reflective property of ellipse (whispering galleries)]] - [[Reflective property of hyperbola]] - [[Fermat's principle and shortest path]] - [[Applications of Conics in Engineering]] ## 🖼️ Concept Map ```mermaid flowchart TD DEF[Focus-directrix definition PF equals PM] -->|foundation for| PROP[Reflective property] PARAB[Parabola y2 = 4ax] -->|has| FOCUS[Focus F at a,0] PARAB -->|differentiate implicitly| TAN[Tangent slope 2a over y1] INRAY[Ray parallel to axis, slope 0] -->|makes angle alpha| TAN FOCRAD[Focal radius PF] -->|makes angle beta| TAN DEF -->|used to simplify| BISECT[Tangent bisects angle] TAN -->|tan alpha equals tan beta| BISECT INRAY -->|reflects through| FOCUS BISECT -->|proves| PROP PROP -->|concentrates rays at| FOCUS PROP -->|explains| APPS[Telescopes, dishes, antennas] FOCUS -->|reverse gives| BEAM[Parallel searchlight beam] ```