Definition via focus, directrix, eccentricity e

3.4.1 · Maths › Conic Sections

WHAT is a conic (focus–directrix definition)

Yeh names/ranges kyun? Number $e$ measure karta hai "tum line se kitna zyada focus ke paas rehte ho."

Agar $e < 1$ hai, point ko hamesha focus ke paas rehna padta hai → curve wrap hokar band ho jaata hai (ellipse).
Agar $e = 1$ hai, dooiyan equal hoti hain → curve band hone se bahut kam chook jaata hai (parabola).
Agar $e > 1$ hai, point line se zyada door focus se ho sakta hai → curve do branches mein khul jaata hai (hyperbola).

HOW to build the equation from scratch (derivation)

Hum general conic ko seedha definition se derive karte hain — kuch bhi memorise nahi.

Setup. Focus ko $F = (0, 0)$ par rakho aur directrix ko vertical line $x = - d$ maano (jahan $d > 0$ ).

Ek point $P = (x, y)$ ke liye:

Focus tak doori: $P F = x^{2} + y^{2}$ . Kyun? Simple distance formula.
Line $x = - d$ tak perpendicular doori: $P M = ∣ x + d ∣$ . Kyun? $x$ aur $- d$ ke beech ka horizontal gap.

Definition $P F = e \cdot P M$ apply karo: $x^{2} + y^{2} = e ∣ x + d ∣$

Dono sides square karo (root aur absolute value dono khatam): $x^{2} + y^{2} = e^{2} (x + d)^{2}$

Har case padho — shapes kyun alag hain. $x^{2}$ ka coefficient $(1 - e^{2})$ dekho:

$e = 1$ : $x^{2}$ term khatam ho jaata hai. Bacha $y^{2} = 2 d x + d^{2}$ — $x$ ki single power → parabola.
$e < 1$ : $(1 - e^{2}) > 0$ , dono squared terms positive hain → bounded, closed → ellipse.
$e > 1$ : $(1 - e^{2}) < 0$ , $x^{2}$ aur $y^{2}$ terms ke opposite signs hain → do open branches → hyperbola.

Parabola (sabse clean case, $e = 1$ )

$x^{2} + y^{2} = (x + d)^{2}$ mein $e = 1$ set karo: $x^{2} + y^{2} = x^{2} + 2 d x + d^{2} \Rightarrow y^{2} = 2 d x + d^{2}$ Agar hum focus ko $(a, 0)$ par aur directrix $x = - a$ par rakhein, toh same steps se standard milta hai $y^{2} = 4 a x$ $4 a$ kyun? Kyunki focus-to-vertex $= a$ aur directrix-to-vertex $= a$ hota hai; algebra inhe $4 a$ mein bundle karta hai.

Figure — Definition via focus, directrix, eccentricity e

Eccentricity as a measured ratio

Ellipse/hyperbola ke liye ek equivalent geometric formula hai: $e = \frac{c}{a}$ jahan $a$ = semi-major axis (centre se vertex tak doori) aur $c$ = centre se focus tak doori. Consistent kyun hai? Dono same "focus ke paas hone vs. curve ki spread" ko encode karte hain.

Worked examples

Common mistakes (steel-manned)

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

Socho zameen par ek dot hai (focus) aur ek seedhi painted line hai (directrix). Tum ghoom ke patthar tabhi giraate ho jahan focus tak tumhare kadam, line tak tumhare kadam se ek certain number of times hoon. Agar woh number chota ho (jaise $\frac{1}{2}$ ), toh patthar ek sundar oval banaate hain. Agar exactly $1$ ho, toh ek U-shape banti hai jo kabhi band nahi hoti. Agar bada ho (jaise $2$ ), toh do curves alag-alag ud jaati hain. Woh ek number hi eccentricity hai — yeh curve ki "personality" hai.

Recall check

Focus–directrix terms mein conic ko kya define karta hai?

Un points

P

ka locus jahan

P F / P M = e

constant ho, jahan

F

focus hai, directrix line hai, aur

P M

us tak perpendicular doori hai.

Eccentricity

e

kya hai?

Fixed ratio (focus tak doori)/(directrix tak perpendicular doori); ek shape number.

Parabola ke liye

e

ki value?

e = 1

(focus aur directrix tak dooiyan equal hoti hain).

Ellipse ke liye

e

ki range?

0 < e < 1

Hyperbola ke liye

e

ki range?

e > 1

(x, y)

se line

x = - d

tak perpendicular doori?

∣ x + d ∣

e = 1

se single-power equation kyun milti hai?

(1 - e^{2}) x^{2}

term zero ho jaata hai, aur

y^{2}

x

mein linear rah jaata hai.

Ellipse/hyperbola mein

e

ka alternative formula?

e = c / a

(focus doori over semi-axis).

Kya bada

e

matlab bada curve hota hai?

Nahi —

e

shape control karta hai, size nahi.

Connections

Parabola - Standard equation y^2=4ax
Ellipse - Standard form and c^2 = a^2 - b^2
Hyperbola - Standard form and c^2 = a^2 + b^2
Circle as limiting case e=0
Distance formula and distance from a point to a line
Latus rectum and semi-latus rectum