Slope, line equations, distance formula aur section formula β sab ek jagah!
π 1. Slope aur Inclination
Ek line ka slope batata hai ki woh kitni tezi se upar/neeche jaati hai. Ramp, road, graph β sab jagah slope use hota hai.
Slope Formula:
m = (yβ β yβ) / (xβ β xβ) = Rise / Run = tan ΞΈ
Jahan ΞΈ = angle line aur x-axis ke beech (inclination angle)
m > 0 β line upar jaati hai (left se right)
m < 0 β line neeche jaati hai
m = 0 β horizontal line
m undefined β vertical line (x = constant)
Example: Slope Nikaalte Hain
Do points A(2, 3) aur B(6, 11) ke beech slope nikalo. Inclination angle bhi batao.
1
m = (yββyβ)/(xββxβ) = (11β3)/(6β2) = 8/4 = 2
2
m = tan ΞΈ = 2 β ΞΈ = tanβ»ΒΉ(2) β 63.43Β°
Parallel aur Perpendicular Lines
Parallel lines: mβ = mβ (same slope, different y-intercept)
Perpendicular lines: mβ Γ mβ = β1 (slopes ka product = β1)
ya: mβ = β1/mβ (negative reciprocal)
Example: Line 1 ka slope = 3/4
β Parallel line ka slope = 3/4
β Perpendicular line ka slope = β4/3
π‘ Angle between two lines: tan Ξ± = |(mβ β mβ)/(1 + mβmβ)| β agar answer = 0, lines parallel; agar denominator = 0, lines perpendicular!
π 2. Straight Line ke Equations
Ek line ko describe karne ke 4 main forms hain. Sab equivalent hain β situation ke hisaab se sahi form use karo.
1. Slope-Intercept Form
y = mx + c
m = slope, c = y-intercept (jahan line y-axis ko cut karti hai). Sabse zyada use hone wali form!
2. Point-Slope Form
y β yβ = m(x β xβ)
Jab ek point aur slope diya ho. (xβ,yβ) us point ke coordinates hain.
Jab do points diye hoon. Slope nikaalte hain phir point-slope use karte hain.
4. Intercept Form
x/a + y/b = 1
a = x-intercept, b = y-intercept. Jab dono intercepts diye hon.
General Form
General Form: ax + by + c = 0
Slope = βa/b
x-intercept = βc/a (y=0 rakho)
y-intercept = βc/b (x=0 rakho)
Kisi bhi form ko general form mein convert kar sakte hain!
Example 1: Slope-Intercept se General Form
Line y = 3x β 5 ko general form mein likho. Dono intercepts nikalo.
1
General form: 3x β y β 5 = 0 (a=3, b=β1, c=β5)
2
x-intercept: y=0 β 3x=5 β x = 5/3
3
y-intercept: x=0 β βy=5 β y = β5 (yaani c = β5)
Example 2: Two Points se Line Equation
Points P(1, 2) aur Q(4, 8) se guzarne wali line ki equation nikalo.
1
m = (8β2)/(4β1) = 6/3 = 2
2
Point-slope: y β 2 = 2(x β 1)
3
y β 2 = 2x β 2 β y = 2x (line origin se guzarti hai!)
Do Lines ka Intersection
Do lines ka intersection: simultaneous equations solve karo
L1: aβx + bβy + cβ = 0
L2: aβx + bβy + cβ = 0
Cramer's rule ya substitution se solve karo.
Example: x + y = 5 aur 2x β y = 1
β Add: 3x = 6 β x = 2, y = 3
β Intersection point: (2, 3)
π 3. Distance aur Section Formula
Distance Formula
Do points P(xβ,yβ) aur Q(xβ,yβ) ke beech:
d = β[(xββxβ)Β² + (yββyβ)Β²]
(Pythagoras theorem ka use! Horizontal aur vertical distances ka hypotenuse)
Example: Distance Nikaalte Hain
A(3, β4) aur B(β1, 2) ke beech distance nikalo.
1
d = β[(-1-3)Β² + (2-(-4))Β²]
2
= β[(-4)Β² + (6)Β²] = β[16 + 36] = β52
3
= 2β13 β 7.21 units
Section Formula
Point P(x,y) jo A(xβ,yβ) aur B(xβ,yβ) ko m:n mein divide karta hai:
Internal Division:
x = (mxβ + nxβ)/(m+n)
y = (myβ + nyβ)/(m+n)
External Division:
x = (mxβ β nxβ)/(mβn)
y = (myβ β nyβ)/(mβn)
Midpoint (m=n=1):
x = (xβ+xβ)/2, y = (yβ+yβ)/2
Example: Section Formula
A(2,5) aur B(8,β1) ko join karne wali line ko point P, 2:1 mein internally divide karta hai. P nikalo.
Point P(xβ,yβ) se line ax + by + c = 0 ki perpendicular distance:
d = |axβ + byβ + c| / β(aΒ² + bΒ²)
Example: Point (3,4) se line 3x β 4y + 5 = 0 ki distance:
d = |3Γ3 β 4Γ4 + 5| / β(9+16) = |9β16+5| / 5 = |β2| / 5 = 2/5 = 0.4 units