Equations of a line — slope-intercept, point-slope, two-point, standard (ax+by+c=0)

2.3.6 · Maths › Coordinate Geometry

WHY: hum line ki equation kyun chahte hain?

Ek line infinite points ka set hoti hai. Hum sab list nahi kar sakte. Toh hum ek condition dhoondते hain jo point $(x, y)$ satisfy karta hai tab aur sirf tab jab woh line par ho. Wahi condition is the equation.

Woh key fact jis par sab forms tike hain:

1. Point–slope form (sab forms ka parent)

WHAT: ek known point $(x_{1}, y_{1})$ aur slope $m$ wali line ki equation.

HOW (scratch se derive karein): Line par koi bhi doosra point $(x, y)$ lo. Slope ki definition ke hisaab se, $(x_{1}, y_{1})$ aur $(x, y)$ ke beech nikala hua slope $m$ ke barabar hona chahiye:

$\frac{y - y _{1}}{x - x _{1}} = m$

Dono sides ko $(x - x_{1})$ se multiply karo (allowed hai, kyunki phir hum point $(x_{1}, y_{1})$ khud bhi include kar lete hain):

Yeh step kyun? Multiply karne se denominator clear ho jaata hai, isliye point $(x_{1}, y_{1})$ (jo fraction ko $0/0$ bana deta) ab legally included ho jaata hai.

2. Slope–intercept form

WHAT: slope $m$ aur woh jagah jahan line $y$ -axis ko cross karti hai (the $y$ -intercept $c$ ).

HOW: $y$ -intercept woh point hai $(0, c)$ . Point–slope mein $(x_{1}, y_{1}) = (0, c)$ plug karo: $y - c = m (x - 0) \Rightarrow y - c = m x$

Yeh step kyun? $(0, c)$ ko known point lene se $x_{1} = 0$ vanish ho jaata hai, aur $y$ isolate ho jaata hai.

3. Two-point form

WHAT: do known points $(x_{1}, y_{1})$ aur $(x_{2}, y_{2})$ se guzarti line.

HOW: Pehle dono points se slope nikalo, $m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}$ , phir point–slope mein substitute karo:

$y - y_{1} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} (x - x_{1})$

Yeh step kyun? Do points ek line puri tarah determine kar dete hain — toh hum bas missing $m$ compute karte hain aur parent form reuse karte hain. Kuch naya yaad karne ki zaroorat nahi.

4. Standard (general) form

WHAT: har seedhi line — vertical bhi — ko is tarah likha ja sakta hai: $a x + b y + c = 0, (a, b) \neq = (0, 0) .$

HOW: $y = m x + c^{'}$ se shuru karo (intercept ka naam $c^{'}$ rakho taaki clash na ho). Sab kuch ek taraf lo: $m x - y + c^{'} = 0.$ Yeh $a x + b y + c = 0$ ki shape mein hai jahan $a = m, b = - 1, c = c^{'}$ . Aur jab line vertical hoti hai ( $x = k$ ), toh hum likhte hain $1 \cdot x + 0 \cdot y - k = 0$ — slope–intercept mein yeh impossible hai, lekin yahaan bilkul theek hai.

Yeh step kyun? $y$ ke liye solve karne se yeh slope–intercept mein aa jaati hai, aur $m$ aur $c$ seedha dikh jaate hain.

Figure — Equations of a line — slope-intercept, point-slope, two-point, standard (ax+by+c=0)

Worked examples

Common mistakes (steel-manned)

Convert-between-forms cheat map

Hamare paas hai	Slope–intercept chahiye	Standard chahiye
point + slope	$y - y_{1} = m (x - x_{1})$ mein plug karo, expand karo	expand karo phir ek taraf lo
do points	pehle $m$ nikalo, phir upar wala	same
standard $a x + b y + c = 0$	$y = - \frac{a}{b} x - \frac{c}{b}$	already hai

Recall Feynman: ek 12-saal ke bache ko samjhao

Socho tum ek ramp par chal rahe ho. Har 1 kadam aage tum upar usi amount jaate ho — woh "usi amount" hi slope hai. Agar main tumhe bataaon (a) ramp kahan se shuru hoti hai aur (b) kitni steep hai, toh tum poori ramp draw kar sakte ho. "Line ki equation" ka har form bas mera wahi do cheezein alag tarike se kehna hai. $y = m x + c$ wala kehta hai "height $c$ par shuru karo, har kadam $m$ chadho." $x = 5$ wala ek wall hai — seedhi upar, aage koi chalna nahi, isliye "upar per step" ki baat nahi ho sakti — yahi reason hai ki uske liye alag form chahiye.

Active-recall flashcards

Do points ke beech slope ki definition kya hai?

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}} = tan θ

, rise over run.

(x_{1}, y_{1})

aur slope

m

wali line ka point–slope form?

y - y_{1} = m (x - x_{1})

Slope–intercept form aur har letter ka matlab?

y = m x + c

;

m

=slope,

c

y

-intercept (

x = 0

par value).

Two-point form?

\frac{y - y _{1}}{x - x _{1}} = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

Line ka standard form?

a x + b y + c = 0

jahan

(a, b) \neq = (0, 0)

a x + b y + c = 0

ki slope?

m = - \frac{a}{b}

(zaroori hai

b \neq = 0

a x + b y + c = 0

y

-intercept?

- \frac{c}{b}

a x + b y + c = 0

x

-intercept?

- \frac{c}{a}

(

y = 0

rakho).

(k, y_{0})

se guzarti vertical line ki equation?

x = k

(slope undefined).

(x_{0}, k)

se guzarti horizontal line ki equation?

y = k

(slope

0

Vertical line

y = m x + c

use kyun nahi kar sakti?

Uski slope undefined hoti hai (run

= 0

, division by zero).

Do points se slope nikaalke line kaise likhen?

m = \frac{y _{2} - y _{1}}{x _{2} - x _{1}}

compute karo,

y - y_{1} = m (x - x_{1})

mein substitute karo.

Connections

Slope of a line — woh single number jis par sab forms depend karte hain.
Angle between two lines — $tan θ = \frac{m _{1} - m _{2}}{1 + m _{1} m _{2}}$ use karta hai.
Parallel and perpendicular lines — $m_{1} = m_{2}$ aur $m_{1} m_{2} = - 1$ .
Distance of a point from a line — standard form $a x + b y + c = 0$ use karta hai.
Intercept form of a line — $\frac{x}{a} + \frac{y}{b} = 1$ .
Straight line as a first-degree equation — kyun degree-1 hamesha ek line hoti hai.