poles and polar

Polar and pole

Equation of the polar with respect to a conic

Doubt:xα + yβ = a²

Kese Aya

Concept Pehle Samjho:

"Chord of Contact"ka equation yeh hota hai. Jab aap kisi point T(α, β) se circle ko do tangents khinchte ho, toh dono points of contact (jahan tangent circle ko touch karti hai) ko milane wali line ko Chord of Contact kehte hain.

Ab Hume yeh equation derive karna hai:

xα + yβ = a²

---

Derivation Step-by-Step:

Step 1: Circle ka equation hai:

x^2 + y^2 = a^2

Step 2: Maan lo point T(α, β) se circle par do tangents khinchi gayi hain. Dono tangents ke contact points A(x₁, y₁) aur B(x₂, y₂) hain.

Step 3: Point A(x₁, y₁) par jo tangent hogi, uska equation kya hoga?

Kisi bhi point(x₁, y₁) par circle x² + y² = a² ki tangent ka equation hota hai:

xx_1 + yy_1 = a^2

(Yeh ek standard formula hai, isko yaad rakhna chahiye.)

Isi tarah, point B(x₂, y₂) par tangent ka equation hoga:

xx_2 + yy_2 = a^2

Step 4: Dono tangents point T(α, β) par milti hain.

Matlab,point T(α, β) dono hi tangent lines par lie karta hai.

1. Pehli tangent xx₁ + yy₁ = a² par point T(α, β) ko rakhne par:

α x₁ + β y₁ = a² \quad \text{(Equation 1)}

2. Dusri tangent xx₂ + yy₂ = a² par point T(α, β) ko rakhne par:

α x₂ + β y₂ = a² \quad \text{(Equation 2)}

Step 5: Ab socho!

Equation(1) aur Equation (2) se kya pata chalta hai?

· Equation (1) ke hisab se, point A(x₁, y₁) ek aisi line ke equation αx + βy = a² ko satisfy karta hai.

· Equation (2) ke hisab se, point B(x₂, y₂) bhi usi line αx + βy = a² ko satisfy karta hai.

Step 6: Final Conclusion:

· Point A aur Point B dono hi line αx + βy = a² par hain.

· Lekin A aur B ko milane wali line toh chord of contact AB hi hai.

· Isliye, line αx + βy = a² hi chord of contact AB ka equation hai.

---

TL;DR (Too Long; Didn't Read) in Hinglish:

· Point T(α, β) se do tangents khinchi.

· Dono points of contact par tangent equations xx₁ + yy₁ = a² aur xx₂ + yy₂ = a² thi.

· Point T(α, β) in dono equations ko satisfy karta hai, isse mila αx₁ + βy₁ = a² aur αx₂ + βy₂ = a².

· Iska matlab dono points of contact (x₁,y₁) aur (x₂,y₂) ek hi nayi line αx + βy = a² par hain.

· Yahi nayi line αx + βy = a² Chord of Contact ka equation hai.

Isliye aapne jo equation dekha xα + yβ = a², woh seedha Chord of Contact ka equation hai jo point T(α, β) se derive hota hai.

Doubt: locus ka equation kese

Donu alag-alag cheezein hain, lekin ek dusre se connected hain.

---

Part 1: Chord of Contact ka Equation (Jo tum samjhe)

· Fixed Point: T(α, β) (Maan lo yeh fixed hai)

· Is fixed point se chord of contact ka equation: xα + yβ = a²

· Yeh ek straight line hai jo fixed hai.

Yahan (α, β) FIXED hai, (x,y) VARIABLE hain jo us line par hain.

---

Part 2: Locus Kaise Banta Hai? (Jo confusion hai)

Ab socho... Point T(α, β) ko hum FIXED nahi, BADALTE hue sochte hain.

Scenario:

1. Humare paas ek FIXED point P(x₁, y₁) hai.

2. Hum P se koi bhi chord AB khinchte hain.

3. Phir A aur B par tangents khinchte hain, jo point T(α, β) par milti hain.

4. Ab har alag chord ke liye, T ka position CHANGE karega.

5. Hum T ke saare possible positions ka path (locus) nikalna chahte hain.

Ab dono equations ko jodte hain:

1. Chord of Contact ka Equation (Jab T(α, β) se khincha):

xα + yβ = a²

2. Kyunki yeh chord P(x₁, y₁) se hokar guzarti hai, toh P ko is equation mein daaloge:

x₁α + y₁β = a²

Yeh dekho! Ab kya hua?

· Pehle: xα + yβ = a² → (α, β) fixed, (x,y) variable.

· Ab Naya Equation: x₁α + y₁β = a² → (x₁, y₁) fixed, (α,β) variable.

Matlab ab (α, β) hi variable ban gaye hain! Kyunki har naye chord ke saath T ka position change ho raha hai.

---

Final Locus Equation:

Jab hum (α, β) ki jagah general variable (h, k) ya phir seedhe (x, y) likhte hain, toh milta hai:

x x₁ + y y₁ = a²

Yehi hai polar ka equation!

---

Summary in Hinglish:

1. Chord of Contact ka equation: xα + yβ = a² (Yahan T(α, β) fixed point hai).

2. Jab hum condition lagate hain ki yeh chord ek fixed point P(x₁, y₁) se guzre, toh equation ban jaata hai: x₁α + y₁β = a².

3. Ab (α, β) hi variable ban jaate hain (kyunki T har bar move karta hai).

4. Isliye (α, β) ko general (x, y) maanke final equation mil jaati hai: x x₁ + y y₁ = a².

Toh haan, chord of contact ka equation use karke hi, ek condition lagane se polar (locus) ka equation aa jaata hai!

Doubt cleared

Scene 1: Ek Fixed Point "T" hai

· Maan lo ek point T(α, β) fixed hai.

· Is T se circle par do tangents khinchi.

· Dono tangents ke touch points ko milaye toh line AB banti hai.

· Is line AB ka equation hai: xα + yβ = a²

· Yeh line FIXED hai kyunki T fixed hai.

· Isko "Chord of Contact" kehte hain.

✅ Yeh tum samjh gaye.

---

Scene 2: Ab roles change karte hain

· Maan lo ek aur point P(x₁, y₁) fixed hai.

· Is P se koi chord AB khinchte hain.

· A aur B par tangents khinchte hain, jo point T par milti hain.

· Ab har nayi chord ke saath T ka position CHANGE hota hai.

Yahan twist aa gaya:

Pehle: T fixed tha, chord of contact fixed thi.

Ab: P fixed hai, T move kar raha hai.

---

Magic Connection:

· Jab bhi koi chord P se guzarti hai, aur uske endpoints par tangents khinchi jaati hain, toh woh tangents ek point T par milti hain.

· Har such T ke liye, uski chord of contact P se hokar guzarti hai.

· Kyunki chord of contact ka equation hai: xα + yβ = a²

· Aur yeh line P(x₁, y₁) se guzarti hai, so:

x₁α + y₁β = a²

---

Locus Kaise Bana?

· Ab socho: P(x₁, y₁) fixed hai.

· (α, β) woh point hai jo har bar change ho raha hai (jaise jaise hum different chords lete hain).

· Equation: x₁α + y₁β = a²

· Yeh equation dikha raha hai ki (α, β) kahan kahan ho sakta hai.

· (α, β) ke saare possible points ka set ko "locus" kehte hain.

Final step: (α, β) ko general coordinates (x, y likh do:

x x₁ + y y₁ = a²

Yahi hai POLAR ka equation!

---

Ek aur simple analogy:

· Chord of Contact: Fixed T ki family of chords.

· Polar: Fixed P ki family of T points.

Donu ek dusre ke mirror jaise hain!

Kya ab thoda clear hua?