Circle as degenerate conic (e = 0) — Hinglish notes

3.4.10 · Maths › Conic Sections

KYA ho raha hai

Figure — Circle as degenerate conic (e = 0)

KYU $e = 0$ ek circle deta hai (scratch se derivation)

Subtle part yeh hai: agar $e = \frac{P S}{P M} = 0$ ho, toh $P S = 0 \cdot P M = 0$ , jo force karega ki $P = S$ — yaani sirf ek single point! Toh hum directly $e = 0$ naively plug nahi kar sakte. Hum ek careful limit lete hain.

Formula Derivation via limit

$e \to 0$ Step 1 — Ellipse set up karo. Ek ellipse ke foci $(\pm c, 0)$ par hote hain, semi-major axis $a$ hoti hai, aur $e = \frac{c}{a}, b^{2} = a^{2} (1 - e^{2}) .$ Yeh step kyun? Hum circle tak ellipse family ke through pahunchte hain, kyunki ellipse woh conic hai jo circle ke "bilkul paas" hai.

Step 2 — $a$ fixed rakh ke $e \to 0$ bhejo. Tab $c = a e \to 0 (dono foci centre mein merge ho jaate hain)$ $b^{2} = a^{2} (1 - e^{2}) \to a^{2} \Rightarrow b \to a .$ Yeh step kyun? Jab $e = 0$ hota hai toh dono foci ek point (centre) par collapse ho jaate hain aur dono axes equal ho jaate hain, $a = b$ . Stretched ellipse relax hokar perfectly round shape ban jaati hai.

Step 3 — Directrix infinity par chali jaati hai. Ellipse ki directrix $x = \frac{a}{e}$ par hoti hai. Jaise $e \to 0$ , $x = \frac{a}{e} \to \infty.$ Yeh step kyun? Directrix "infinity par chali jaati hai," aur exactly isili wajah se circle ki koi directrix nahi hoti — $e = 0$ ke consistent hai.

Conclusion. Ellipse ka equation $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ , jab $b = a$ ho, ban jaata hai: $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{a ^{2}} = 1 ⟹ x^{2} + y^{2} = a^{2} .$

KAISE use karte hain — worked examples

Forecast-then-Verify

Common mistakes (Steel-manned)

Feynman

Recall Ek 12-saal ke bachche ko samjhao

Socho ek ball ko ek string se ek jagah pin kar ke ghuma rahe ho — tum ek perfectly round circle banate ho, aur koi fark nahi padta tum kis taraf ghuma rahe ho, woh hamesha same dikhti hai. Doosri shapes jaise ovals (ellipses) ka ek "lamba rasta" aur ek "chhota rasta" hota hai. Number $e$ measure karta hai shape kitni squashed hai. Circle bilkul squashed nahi hoti, isliye uska $e$ $0$ hota hai. Yeh "sabse round possible" oval hai.

Flashcards

Circle ki eccentricity kya hoti hai?

e = 0

Hum directly

P S / P M = 0

set karke circle kyun nahi pa sakte?

Yeh

P S = 0

force kar dega, ek single point dega; instead ellipse ka limit

e \to 0

lo.

Jab ellipse mein

e \to 0

hota hai, toh dono foci ka kya hota hai?

Woh ek single point mein merge ho jaate hain — centre.

Jaise

e \to 0

hota hai, directrix kahan jaati hai?

Infinity par (

x = a / e \to \infty

), isliye circle ki koi directrix nahi hoti.

Ellipse ke liye

a

b

e

ka relation?

b^{2} = a^{2} (1 - e^{2})

b^{2} = a^{2} (1 - e^{2})

mein

e = 0

rakhne par kya milta hai?

b = a

(equal axes) → ek circle.

Circle ka standard equation, centre

(h, k)

, radius

r

(x - h)^{2} + (y - k)^{2} = r^{2}

Circle ki general form?

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

General form se centre aur radius?

Centre

(- g, - f)

, radius

g^{2} + f^{2} - c

x^{2} + y^{2} + 2 g x + 2 f y + c = 0

point circle kab hota hai?

Jab

g^{2} + f^{2} - c = 0

ho (radius 0).

(0, 0), (2, 0), (0, 2)

se guzarne wale circle ka centre?

(1, 1)

, radius

2

Connections

Conic Sections — parent family, ek plane dwara cone ko kaatne se.
Eccentricity — single parameter $e$ jo sabhi conics classify karta hai.
Ellipse — circle uska $e \to 0$ limit hai ( $a = b$ ).
Parabola ( $e = 1$ ) aur Hyperbola ( $e > 1$ ) — baaki conic members.
Distance Formula — woh tool jo circle equation build karta hai.
Completing the Square — general form aur standard form ke beech convert karta hai.
Degenerate Conics — point circle, single point, pair of lines: limiting cases.