Sum of focal radii = 2a property — Hinglish notes

3.4.5 · Maths › Conic Sections

80/20 core idea: Kisi bhi ellipse ke kisi bhi point ke liye, dono foci ki taraf ki distances hamesha $2 a$ mein add hoti hain (major axis ki length). Yahi ek fact ellipse ki definition hai.

The Big Picture

Figure — Sum of focal radii = 2a property

Derivation From Scratch (kuch bhi memorise karne ki zarurat nahi!)

Hum standard ellipse use karte hain $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b > 0.$ Iske foci $S (a e, 0)$ aur $S^{'} (- a e, 0)$ hain, jahaan $e$ eccentricity hai aur $b^{2} = a^{2} (1 - e^{2})$ .

Step 1 — Distance formula se ek focal radius likho. Maano $P = (x, y)$ ellipse par lie karta hai. $r_{1} = S P = (x - a e)^{2} + y^{2} .$ Yeh step kyun? Humein distances chahiye, aur distance = coordinate differences ke squares ka square-root hota hai.

Step 2 — Ellipse equation use karke $y^{2}$ ko hatao. Equation se, $y^{2} = b^{2} (1 - \frac{x ^{2}}{a ^{2}})$ . $b^{2} = a^{2} - a^{2} e^{2}$ substitute karo: $y^{2} = (a^{2} - a^{2} e^{2}) (1 - \frac{x ^{2}}{a ^{2}}) = a^{2} - a^{2} e^{2} - x^{2} + e^{2} x^{2} .$ Yeh step kyun? Hum $r_{1}$ ko sirf $x$ ke terms mein likhna chahte hain, taaki sum cleanly simplify ho jaye.

Step 3 — Square root ke andar expand karo. $(x - a e)^{2} + y^{2} = x^{2} - 2 a e x + a^{2} e^{2} + a^{2} - a^{2} e^{2} - x^{2} + e^{2} x^{2} .$ $x^{2}$ , $a^{2} e^{2}$ terms cancel ho jaate hain: $= a^{2} - 2 a e x + e^{2} x^{2} = (a - e x)^{2} .$ Yeh step kyun? $b^{2} = a^{2} (1 - e^{2})$ use karne ka poora point yahi hai ki yeh ek perfect square mein collapse ho jaata hai.

Step 4 — Square root lo. $r_{1} = (a - e x)^{2} = a - e x .$ (Ellipse par kisi point ke liye $- a \leq x \leq a$ aur $0 < e < 1$ , isliye $a - e x > 0$ ; root positive hai.)

Step 5 — Doosre focus $S^{'} (- a e, 0)$ ke liye bhi yahi karo. $- a e$ ki jagah $+ a e$ use karke identical algebra se: $r_{2} = S^{'} P = a + e x .$

Step 6 — Unhe add karo. $r_{1} + r_{2} = (a - e x) + (a + e x) = 2 a . ■$

Worked Examples

Common Mistakes (Steel-manned)

Active Recall

Recall Apne aap ko test karo (answer karne ke baad reveal karo)

Ellipse par kisi bhi point ke liye $S P + S^{'} P$ kya hai? → $2 a$ .
Near (right) focus se focal radius? → $a - e x$ .
Sum $x$ par depend kyun nahi karta? → $- e x$ aur $+ e x$ cancel ho jaate hain.
Constant kaun si axis length hai? → major axis, $2 a$ .
Ellipse vs hyperbola constant? → sum vs difference.

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho ek board mein do nails hain aur unke around ek loose loop of string hai jisme ek pencil tight hai. Jab tum draw karte ho, pencil ek squashed circle (ek ellipse) banati hai. String ki length kabhi nahi badlti — isliye nail A se pencil ki distance plus nail B se pencil ki distance hamesha ek hi number hai. Woh fixed number oval ki width hai apni longest direction mein, jise hum $2 a$ kehte hain. Nails "foci" hain.

Connections

Ellipse - Standard Equation — jahaan se $a, b, e$ aate hain.
Eccentricity of Conics — $a \pm e x$ ke andar ka $e$ .
Directrix and Focal Distance — focal radius $= e \cdot$ (distance to directrix).
Hyperbola - Difference of Focal Radii = 2a — "difference" wala cousin.
Definition of Conic as Locus — ellipse as constant-sum locus.

Ellipse par ek point ke liye

S P + S^{'} P

kya hota hai?

Major axis ki length,

2 a

Right focus

S (a e, 0)

se focal radius kya hai?

S P = a - e x

Left focus

S^{'} (- a e, 0)

se focal radius kya hai?

S^{'} P = a + e x

Focal radii ka sum constant kyun hai?

- e x

aur

+ e x

terms cancel ho jaate hain,

2 a

position se independent rehta hai.

Derivation mein,

(x - a e)^{2} + y^{2}

kis cheez mein simplify hota hai?

Perfect square

(a - e x)^{2}

mein.

Algebra ko perfect square mein collapse karne wala substitution kaun sa hai?

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

Ellipse constant vs hyperbola constant?

Ellipse: sum

r_{1} + r_{2} = 2 a

; hyperbola: difference

∣ r_{1} - r_{2} ∣ = 2 a

Foci

(\pm 3, 0)

aur focal-radii sum

10

diya ho toh

a

aur

b^{2}

nikalo.

2 a = 10 \Rightarrow a = 5

;

a e = 3 \Rightarrow e = 3/5

;

b^{2} = 25 (1 - 9/25) = 16

S^{'} P - S P

x

ke terms mein kya hoga?

2 e x