Parabola — standard forms (4 orientations), focus, directrix, latus rectum, axis

1. Definition (first principles)

WHAT are the parts?

Focus $F$ — the special point.
Directrix — the special line.
Axis — the line through the focus, perpendicular to the directrix (the axis of symmetry).
Vertex — the point on the parabola on the axis, exactly midway between focus and directrix.
Latus rectum — the chord through the focus, parallel to the directrix.

2. Deriving the standard form $y^2 = 4ax$

HOW — set up convenient coordinates. Put the vertex at the origin, axis along the $x$ -axis, focus to the right at $F=(a,0)$ with $a>0$ . Since the vertex is midway between focus and directrix, the directrix is $x=-a$ .

Let $P=(x,y)$ .

Distance to focus: $PF=\sqrt{(x-a)^2+y^2}$ .
Distance to directrix $x=-a$ : perpendicular distance $=|x+a|$ .

Set them equal (the definition), then square: $(x-a)^2+y^2=(x+a)^2$

Why square? To kill the square root and the absolute value cleanly — both sides are non-negative.

Expand: $x^2-2ax+a^2+y^2 = x^2+2ax+a^2$

The $x^2$ and $a^2$ cancel: $y^2 = 4ax$

3. Latus rectum — derive its length

WHY care? It tells you how "wide" the parabola is at the focus — a quick sketch aid.

The latus rectum is the chord through $F=(a,0)$ parallel to the directrix, i.e. the vertical line $x=a$ . Substitute into $y^2=4ax$ : $y^2=4a\cdot a=4a^2 \implies y=\pm 2a.$

So the endpoints are $(a,2a)$ and $(a,-2a)$ . Length: $\text{LR} = 2a-(-2a) = 4a.$

4. The four orientations

By swapping signs and swapping $x\leftrightarrow y$ in the same derivation, we get all four. (Always $a>0$ ; the sign in the equation controls direction.)

Figure — Parabola — standard forms (4 orientations), focus, directrix, latus rectum, axis

Equation	Opens	Focus	Directrix	Axis	LR
$y^2=4ax$	Right	$(a,0)$	$x=-a$	$y=0$	$4a$
$y^2=-4ax$	Left	$(-a,0)$	$x=a$	$y=0$	$4a$
$x^2=4ay$	Up	$(0,a)$	$y=-a$	$x=0$	$4a$
$x^2=-4ay$	Down	$(0,-a)$	$y=a$	$x=0$	$4a$

5. Worked examples

6. Common mistakes (steel-manned)

7. Active recall

Recall Quick self-test (hide answers, forecast first!)

What single property defines a parabola? → equidistant from focus and directrix.
Where is the vertex relative to focus & directrix? → midpoint.
$y^2=4ax$ : focus? directrix? → $(a,0)$ , $x=-a$ .
Length of latus rectum? → $4a$ .
$x^2=-4ay$ opens where? → down.

Recall Feynman: explain to a 12-year-old

Imagine a point (the focus) and a straight fence (the directrix) on the ground. You walk so that you're always exactly as far from the point as from the fence. The curved path you trace is a parabola — like a satellite dish or the path of a thrown ball. The closer the point is to the fence, the tighter the curve; the farther apart, the wider it flares. The "vertex" is where you're closest to both — right in the middle between them.

Flashcards

Defining property of a parabola

Every point is equidistant from the focus and the directrix (

e=1

Standard form opening right

y^2=4ax

with

a>0

Focus and directrix of

y^2=4ax

Focus

(a,0)

, directrix

x=-a

Latus rectum length of

y^2=4ax

4a

Endpoints of latus rectum of

y^2=4ax

(a,\,2a)

and

(a,\,-2a)

How to get

a

from

y^2=12x

4a=12\Rightarrow a=3

(divide coefficient by 4).

Which axis is the axis of symmetry if equation has

x^2

The

y

-axis (opens up/down).

Direction of

x^2=-4ay

Opens downward; focus

(0,-a)

, directrix

y=a

Where is the vertex located

Midpoint between focus and directrix, on the axis.

Equation from focus

(0,4)

, directrix

y=-4

x^2=16y

Connections

Conic Sections — Overview (parabola is the $e=1$ conic)
Ellipse — standard forms and Hyperbola — standard forms (compare $e<1$ , $e>1$ )
Eccentricity and focus-directrix definition
Tangents and Normals to a Parabola
Coordinate Geometry — Distance Formula (the tool used in the derivation)
Projectile Motion (physics application of parabolic paths)

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Parabola ka ek hi core idea hai: har point aise chalta hai ki uski focus (ek fixed point) se doori aur directrix (ek fixed line) se doori exactly barabar rahe. Yehi ek line yaad rakh lo, baaki sab formula khud ban jaayega. Isko coordinates mein likho — $\sqrt{(x-a)^2+y^2}=|x+a|$ — dono side square karo, cancel karo, aur seedha $y^2=4ax$ aa jaata hai. Koi ratta nahi.

Char orientations sirf sign aur variable swap se aate hain. Agar equation mein $y^2$ hai to axis $x$ -axis pe hai (right/left khulta hai); agar $x^2$ hai to axis $y$ -axis pe (up/down khulta hai). Sign positive ho to focus wali side khulega, negative ho to ulti side. Focus $(a,0)$ type hota hai aur directrix uske opposite side $x=-a$ — kyunki vertex dono ke beech midpoint hai.

Latus rectum ka matlab hai focus se guzarti hui wo chord jo directrix ke parallel hai. $x=a$ daal ke dekho $y=\pm 2a$ , to length $=4a$ . Isiliye equation mein coefficient $4a$ likhte hain — taaki turant latus rectum padha ja sake. Exam trick: $y^2=12x$ mein $4a=12$ , matlab $a=3$ , coefficient ko 4 se divide karo, direct $a$ mat samajh lena.

Do sabse common galtiyan: (1) coefficient ko $a$ maan lena — nahi, wo $4a$ hai. (2) directrix ko focus wali side rakh dena — nahi, wo opposite side pe hoti hai. In dono ko pakad lo to parabola ke saare questions clear ho jaayenge.

Parabola — standard forms (4 orientations), focus, directrix, latus rectum, axis

1. Definition (first principles)

2. Deriving the standard form $y^2 = 4ax$

3. Latus rectum — derive its length

4. The four orientations

5. Worked examples

6. Common mistakes (steel-manned)

7. Active recall

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Conic Sections

Connections