Ellipse — standard forms, semi-major - minor axes, foci, eccentricity, latus rectum

3.4.4 · Maths › Conic Sections

1. Ellipse KYA hai (defining property)

WHY constant ko $2 a$ kehte hain: hum dekhenge ki yeh constant major axis ki poori length ke barabar hoti hai, aur major axis ki half-length $a$ hai. Isse $2 a$ naam dene se baad mein algebra asaan ho jaati hai.

2. Standard equation KAISE derive karte hain (scratch se)

Foci ko $x$ -axis par symmetrically rakhte hain: $F_{1} = (- c, 0), F_{2} = (c, 0), c > 0.$

Maano $P = (x, y)$ . Defining property kehti hai: $(x + c)^{2} + y^{2} + (x - c)^{2} + y^{2} = 2 a .$

Step 1 — ek root ko isolate karo. Kyun? Do square roots ek saath square karke khatam nahi ho sakte; isolate karo taaki ek khatam ho. $(x + c)^{2} + y^{2} = 2 a - (x - c)^{2} + y^{2} .$

Step 2 — dono sides ko square karo. $(x + c)^{2} + y^{2} = 4 a^{2} - 4 a (x - c)^{2} + y^{2} + (x - c)^{2} + y^{2} .$

Step 3 — expand karo aur cancel karo. Kyun? $y^{2}$ aur zyaadatar $x$ -terms cancel ho jaate hain, bacha hua root akela reh jaata hai. $x^{2} + 2 c x + c^{2} = 4 a^{2} - 4 a (x - c)^{2} + y^{2} + x^{2} - 2 c x + c^{2} .$ $4 c x = 4 a^{2} - 4 a (x - c)^{2} + y^{2} .$ $a (x - c)^{2} + y^{2} = a^{2} - c x .$

Step 4 — phir se square karo. $a^{2} [(x - c)^{2} + y^{2}] = a^{4} - 2 a^{2} c x + c^{2} x^{2} .$ $a^{2} x^{2} - 2 a^{2} c x + a^{2} c^{2} + a^{2} y^{2} = a^{4} - 2 a^{2} c x + c^{2} x^{2} .$

$- 2 a^{2} c x$ cancel karo: $a^{2} x^{2} + a^{2} c^{2} + a^{2} y^{2} = a^{4} + c^{2} x^{2} .$ $(a^{2} - c^{2}) x^{2} + a^{2} y^{2} = a^{2} (a^{2} - c^{2}) .$

Step 5 — $b^{2} = a^{2} - c^{2}$ define karo. Yeh positive kyun hai? Kyunki $2 a$ (string) $2 c$ (nail separation) se lambi hai, isliye $a > c$ , isliye $a^{2} - c^{2} > 0$ . Isse $b^{2}$ kehte hain: $b^{2} x^{2} + a^{2} y^{2} = a^{2} b^{2} .$

$a^{2} b^{2}$ se divide karo:

3. Equation se geometry padhna

Figure — Ellipse — standard forms, semi-major - minor axes, foci, eccentricity, latus rectum

$y = 0$ set karo: $x = \pm a$ → vertices $(\pm a, 0)$ . $x = 0$ set karo: $y = \pm b$ → co-vertices $(0, \pm b)$ .

Latus rectum derive karna. Hum kyun care karte hain? Yeh ellipse ki width hai ek focus ke across — ek quick "size" gauge. $x = c$ ko equation mein daalo: $\frac{c ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 \Rightarrow y^{2} = b^{2} (1 - \frac{c ^{2}}{a ^{2}}) = b^{2} \cdot \frac{a ^{2} - c ^{2}}{a ^{2}} = \frac{b ^{2} \cdot b ^{2}}{a ^{2}} = \frac{b ^{4}}{a ^{2}} .$ Toh $y = \pm b^{2} / a$ , aur full chord length hai:

4. DOOSRA standard form (major axis vertical)

Agar bada denominator $y^{2}$ ke neeche hai: $\frac{x ^{2}}{b ^{2}} + \frac{y ^{2}}{a ^{2}} = 1, a > b > 0.$ Ab major axis $y$ -axis ke along hai, foci $(0, \pm c)$ par hain, $c = a^{2} - b^{2}$ .

5. Worked examples

Recall Feynman: 12-saal ke bachche ko samjhao

Ek board mein do nails ठokو aur unpar string ka loop daalo. Loop ko pencil ki tip se tight khincho aur pencil ko poori taraf ghumaao — jo shape banti hai woh ellipse hai. Kyunki string ki length kabhi nahi badlti, do nail-se-pencil distances hamesha ek hi number mein add hote hain. Nails hain foci. Sabse lamba rasta major axis hai (uska aadha $a$ hai), sabse chhota minor axis hai (aadha $b$ hai). Agar nails ek hi jagah rakh do, toh sirf ek perfect circle banta hai — isliye circle ek "lazy ellipse" hai jiska eccentricity zero hai. Nails jitna ends ki taraf jaate hain, oval utna hi stretched aur patla hota hai — bada eccentricity.

Active-recall flashcards

Ellipse ki defining focal property kya hai?

Kisi bhi point se do foci tak distances ka sum constant hota hai, jo

2 a

ke barabar hai.

Ellipse ke liye

a

b

c

ke beech relation?

b^{2} = a^{2} - c^{2}

, equivalently

c^{2} = a^{2} - b^{2}

(toh

a^{2} = b^{2} + c^{2}

Ellipse ki eccentricity ka formula aur uski range?

e = c / a = a^{2} - b^{2} / a

, jahaan

0 \leq e < 1

Latus rectum ki length?

2 b^{2} / a

\frac{x ^{2}}{p} + \frac{y ^{2}}{q} = 1

se major axis kaun sa hai yeh kaise pata karte hain?

Bada denominator major-axis variable ke neeche hota hai;

a^{2} = max (p, q)

\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1

mein

a > b

ke liye foci ke coordinates?

(\pm c, 0)

jahaan

c = a^{2} - b^{2}

e = 0

wala ellipse kaunsi shape hoti hai?

Circle (foci centre par merge ho jaate hain).

\frac{x ^{2}}{9} + \frac{y ^{2}}{25} = 1

ke liye foci kahaan hain?

y

-axis par

(0, \pm 4)

par, kyunki

a^{2} = 25, b^{2} = 9, c = 4

Connections

Conic Sections — overview (ellipse as a section of a cone with plane angle between base and slant)
Circle (special case $e = 0$ , $a = b$ )
Hyperbola — standard forms (uses difference of focal distances; there $c^{2} = a^{2} + b^{2}$ )
Parabola ( $e = 1$ limiting boundary case)
Eccentricity and directrix (focus–directrix unified definition $P F = e \cdot P d$ )
Pythagorean theorem (the $a^{2} = b^{2} + c^{2}$ triangle)