2.3.8
Do lines tab parallel hoti hain jab woh ek same direction mein point karti hain — chahe unhe kitna bhi extend karo, woh kabhi nahi milti. Ek line ki direction ek number se capture hoti hai: uska slope m same slope " mein (m 1 = m 2 Kyun yeh obvious lagta hai jab dikh jaaye: slope = kitna rise karte ho ek fixed run ke liye. Agar do lines same amount rise karti hain same run ke liye, toh woh bilkul ek jaisi chadh rahi hain, toh unke beech ka vertical gap constant rehta hai → woh kabhi cross nahi karti.
Ek line ke liye jo points ( x 1 , y 1 ) ( x 2 , y 2 ) m = x 2 − x 1 y 2 − y 1 = run rise = tan θ θ positive x ke saath banati hai.
Intuition General-form shortcutEk line a x + b y + c = 0 m = − b a y y = − b a x − b c a 1 x + b 1 y + c 1 = 0 a 2 x + b 2 y + c 2 = 0 − b 1 a 1 = − b 2 a 2 ⟺ a 2 a 1 = b 2 b 1 . a x + b y + k = 0 a , b
Worked example 2 — Ek point se hoti hui line, ek di gayi line ke parallelWoh line dhundho jo ( 1 , 4 ) 3 x + 2 y = 7
Di gayi slope: m = − 2 3 Kyun? m = − a / b a = 3 , b = 2
Parallel ⇒ same slope − 2 3 Kyun? Equal-slopes theorem.
Point–slope: y − 4 = − 2 3 ( x − 1 )
Theek karo: 2 y − 8 = − 3 x + 3 ⇒ 3x + 2y = 11 Shortcut check: 3 x + 2 y = k ( 1 , 4 ) 3 + 8 = 11
Worked example 4 — Do parallels ke beech distance (bonus, samea , b 3 x + 4 y = 10 3 x + 4 y = 20 d = a 2 + b 2 ∣ c 1 − c 2 ∣ = 9 + 16 ∣10 − 20∣ = 5 10 = 2. Formula kyun kaam karta hai: dono lines same normal direction ( a , b )
Common mistake "Equal slopes = same line"Kyun sahi lagta hai: equal slopes jaldi mein identical lagte hain.
Fix: Same slope + different intercept ⇒ parallel (kabhi nahi milti). Same slope aur same intercept ⇒ same line (infinitely many meeting points). Constant bhi hamesha check karo.
Common mistake Vertical lines bhool jaanaKyun sahi lagta hai: rule "m 1 = m 2 Fix: Vertical lines (x = c undefined hota hai. Do verticals parallel hoti hain lekin unke slopes "equal numbers" nahi hain — dono undefined hain. Vertical lines ko inspection se handle karo.
Common mistake Parallel ke liyea 2 a 1 = b 2 b 1 = c 2 c 1 Kyun sahi lagta hai: yeh coincident (same line) condition hai, ek step aage.
Fix: Parallel ke liye chahiye a 2 a 1 = b 2 b 1 = c 2 c 1 c
Recall Quick self-test (answers chupaao, pehle forecast karo)
Slope of 5 x − 2 y + 7 = 0 m = 5/2
Line through ( 0 , 0 ) y = 3 x − 4 y = 3 x
Kya x = 2 x = 9
a x + b y + c = 0 a 2 a 1 = b 2 b 1 = c 2 c 1
Recall Feynman: ek 12-saal ke bacche ko explain karoSocho do escalators side by side hain. Agar dono bilkul same steepness par upar jaate hain, toh woh forever ek doosre ke saath chalte hain aur kabhi nahi takrate — yahi "parallel" hai. "Steepness number" slope hai. Same steepness = parallel. Agar woh same jagah se bhi start karein, toh woh literally ek hi escalator hain. Agar ek zyada steep hai, toh woh eventually cross karenge.
"Same tilt, no meet." Parallel lines ka same tilt (slope) hota hai aur woh kabhi nahi milti. Aur a x + b y + c "keep a , b c
Two lines are parallel iff their slopes satisfy what? m 1 = m 2
Slope of a x + b y + c = 0 m = − a / b
Parallel condition using coefficients of a 1 x + b 1 y + c 1 = 0 a 2 a 1 = b 2 b 1 = c 2 c 1
If slopes AND intercepts are equal, the lines are…? The same (coincident) line, not just parallel.
Why do equal slopes imply the lines never meet? Same rise-per-run keeps the vertical gap constant, so they never intersect.
Form of a line parallel to a x + b y + c = 0 a x + b y + k = 0 a , b k
Slope in terms of angle θ m = tan θ
Are two vertical lines parallel, and what about their slopes? Yes, parallel; both slopes are undefined.
Distance between parallels a x + b y + c 1 = 0 a x + b y + c 2 = 0 equal corresponding angles
equal angles equal tangents
