Definition via focus, directrix, eccentricity e

WHAT is a conic (focus–directrix definition)

WHY these names/ranges? The number $e$ measures "how much closer you must be to the focus than to the line."

If $e<1$ , the point must always stay closer to the focus → the curve wraps around and closes up (ellipse).
If $e=1$ , distances are equal → the curve just barely fails to close (parabola).
If $e>1$ , the point can be farther from the focus than from the line → the curve opens out into two branches (hyperbola).

HOW to build the equation from scratch (derivation)

We derive the general conic from the raw definition — nothing memorised.

Setup. Put the focus at $F=(0,0)$ and let the directrix be the vertical line $x = -d$ (with $d>0$ ).

For a point $P=(x,y)$ :

Distance to focus: $PF = \sqrt{x^2+y^2}$ . Why? Plain distance formula.
Perpendicular distance to line $x=-d$ : $PM = |x+d|$ . Why? Horizontal gap between $x$ and $-d$ .

Apply the definition $PF = e\cdot PM$ : $\sqrt{x^2+y^2} = e\,|x+d|$

Square both sides (kills the root and the absolute value): $x^2 + y^2 = e^2 (x+d)^2$

Read off each case — WHY the shapes differ. Look at the $x^2$ coefficient $(1-e^2)$ :

$e=1$ : the $x^2$ term dies. Left with $y^2 = 2d x + d^2$ — a single power of $x$ → parabola.
$e<1$ : $(1-e^2)>0$ , both squared terms are positive → bounded, closed → ellipse.
$e>1$ : $(1-e^2)<0$ , the $x^2$ and $y^2$ terms have opposite signs → two open branches → hyperbola.

The parabola (cleanest case, $e=1$ )

Set $e=1$ in $x^2+y^2 = (x+d)^2$ : $x^2+y^2 = x^2 + 2dx + d^2 \;\Rightarrow\; y^2 = 2dx + d^2$ If instead we place focus at $(a,0)$ and directrix $x=-a$ , the same steps give the standard $\boxed{y^2 = 4ax}$ Why $4a$ ? Because focus-to-vertex $=a$ and directrix-to-vertex $=a$ ; the algebra bundles them as $4a$ .

Figure — Definition via focus, directrix, eccentricity e

Eccentricity as a measured ratio

For ellipse/hyperbola there's an equivalent geometric formula: $e = \frac{c}{a}$ where $a$ = semi-major axis (distance centre→vertex) and $c$ = distance centre→focus. Why consistent? Both encode the same "closeness to focus vs. spread of curve."

Worked examples

Common mistakes (steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine a dot on the ground (the focus) and a straight painted line (the directrix). You walk around and drop pebbles only where your steps to the dot are exactly a certain number of times your steps to the line. If that number is small (like $\tfrac12$ ), your pebbles make a nice oval. If it's exactly $1$ , they make a U-shape that never closes. If it's big (like $2$ ), they make two curves flying apart. That one number is the eccentricity — it's the "personality" of the curve.

Recall check

What defines a conic in focus–directrix terms?

The locus of points

P

with

PF/PM = e

constant, where

F

is the focus, directrix the line,

PM

the perpendicular distance to it.

What is eccentricity

e

The fixed ratio (distance to focus)/(perpendicular distance to directrix); a shape number.

Value of

e

for a parabola?

e=1

(distances to focus and directrix are equal).

Range of

e

for an ellipse?

0<e<1

Range of

e

for a hyperbola?

e>1

Perpendicular distance from

(x,y)

to line

x=-d

|x+d|

Why does

e=1

give a single-power equation?

The

(1-e^2)x^2

term becomes zero, leaving

y^2

linear in

x

Alternative formula for

e

in ellipse/hyperbola?

e=c/a

(focus distance over semi-axis).

Does larger

e

mean a larger curve?

No —

e

controls shape, not size.

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

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, saare conics — circle, ellipse, parabola, hyperbola — ek hi rule se bante hain. Ek fixed point hota hai jise focus kehte hain, aur ek fixed straight line hoti hai jise directrix kehte hain. Ab kisi bhi point $P$ ke liye do distances lo: focus tak ki distance $PF$ , aur directrix tak ki perpendicular distance $PM$ . Inka ratio $PF/PM$ agar hamesha ek constant number rahe, toh us number ko eccentricity $e$ kehte hain, aur banne wali curve ek conic hoti hai.

Sabse important baat: ye $e$ ek single dial hai jo shape decide karta hai. $e<1$ ho toh ellipse (band, oval jaisa), $e=1$ ho toh parabola (U-shape, band nahi hoti), aur $e>1$ ho toh hyperbola (do alag branches). $e=0$ special case circle hai. Yaad rakho — $e$ size nahi batata, sirf shape batata hai. Chhoti ya badi ellipse dono ka $e$ same ho sakta hai.

Equation nikalne ka trick simple hai: $PF = e\cdot PM$ likho, dono taraf square karo (kyunki dono distances positive hain, isliye safe hai), aur simplify karo. Focus origin par aur directrix $x=-d$ lene se $\sqrt{x^2+y^2}=e|x+d|$ milta hai, square karke $x^2+y^2=e^2(x+d)^2$ . Yahan $x^2$ ka coefficient $(1-e^2)$ hai — isi se pata chalta hai kaunsa conic banega. $e=1$ par ye zero ho jaata hai, isliye parabola ka single-power equation aata hai.

Exam mein galti sirf ek jagah hoti hai: log directrix ki distance bhi $\sqrt{x^2+y^2}$ jaise likh dete hain. Nahi! Line tak ki distance hamesha perpendicular hoti hai, jaise $x=-d$ ke liye $|x+d|$ . Isko sahi le liya toh baaki sab algebra automatic ho jaata hai.

Definition via focus, directrix, eccentricity e

WHAT is a conic (focus–directrix definition)

HOW to build the equation from scratch (derivation)

The parabola (cleanest case, $e=1$ )

Eccentricity as a measured ratio

Worked examples

Common mistakes (steel-manned)

Recall check

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Conic Sections

Connections