Chapter 9 - Straight Lines

Beginner

Topper

1. Introduction - Slope and Inclination m = tanθ

Slope measures steepness m Inclination θ angle line with positive x-axis anticlockwise 0° to 180°. Relation m = tanθ Example θ=45° m=1 slope 1 line rises 45° equally.

Slope from two points P(x1,y1) Q(x2,y2) m=(y2-y1)/(x2-x1) rise over run. Example P(1,2) Q(4,6) m=4/3≈1.33 θ≈53.1°.

Types: Positive m>0 θ acute rises left to right, Negative m<0 θ obtuse falls, Zero m=0 horizontal, Undefined vertical.

Slope steepness m Inclination θ 0° to 180° m=tanθ θ=45° m=1 | Slope two points m=(y2-y1)/(x2-x1) Example P(1,2) Q(4,6) m=4/3 θ≈53.1° | Types Positive m>0 acute rises Negative m<0 obtuse falls Zero m=0 horizontal Undefined vertical

Introduction - Slope Inclination m=tanθ

2. Various Forms - Slope-Intercept Point-Slope Two-Point

Slope-intercept y=mx+c m slope c y-intercept (0,c). Example y=2x+3 slope 2 intercept 3.

Point-slope y-y1=m(x-x1) point (x1,y1) on line Example (2,3) slope 4 y-3=4(x-2) y=4x-5.

Two-point (y-y1)/(y2-y1)=(x-x1)/(x2-x1) through two points Example (1,2) (4,6) (y-2)/4=(x-1)/3 →4x-3y+2=0.

Slope-intercept y=mx+c m slope c y-intercept (0,c) Example y=2x+3 | Point-slope y-y1=m(x-x1) Example (2,3) slope 4 y=4x-5 | Two-point (y-y1)/(y2-y1)=(x-x1)/(x2-x1) Example (1,2) (4,6) →4x-3y+2=0

Forms - Slope-Intercept Point-Slope Two-Point

3. Slope-Intercept and Intercept Forms

Slope-intercept y=mx+c m slope c y-intercept. Example slope 2 intercept 5 y=2x+5.

Intercept form x/a + y/b =1 a x-intercept (a,0) b y-intercept (0,b). Example cuts x-axis (4,0) a=4 y-axis (0,3) b=3 equation x/4+y/3=1 →3x+4y=12.

Converting x/a+y/b=1 → y=(-b/a)x+b slope m=-b/a c=b. Example x/4+y/3=1 slope -3/4 intercept 3.

Slope-intercept y=mx+c | Example slope 2 intercept 5 y=2x+5 | Intercept form x/a + y/b=1 a x-intercept b y-intercept Example cuts (4,0) (0,3) x/4+y/3=1 →3x+4y=12 | Converting y=(-b/a)x+b slope -b/a c=b Example x/4+y/3=1 slope -3/4 intercept 3

Slope-Intercept Intercept Form

4. Normal Form and General Equation Ax+By+C=0

Normal form x cos ω + y sin ω = p p perpendicular distance origin to line p≥0 ω angle between perpendicular origin to line and x-axis. cos²+sin²=1 normalized. Example Convert x/4+y/3=1 p=12/5=2.4 cos=0.6 sin=0.8 x(0.6)+y(0.8)=2.4.

General Ax+By+C=0 A,B not both zero first degree represents line Slope m=-A/B if B≠0 Example 3x+4y-12=0 slope -3/4.

Normal form x cos ω + y sin ω =p p perpendicular distance origin p≥0 ω angle perpendicular and x-axis cos²+sin²=1 | Example x/4+y/3=1 p=2.4 cos 0.6 sin 0.8 normal 0.6x+0.8y=2.4 | General Ax+By+C=0 A,B not both zero first degree line Slope m=-A/B | Example 3x+4y-12=0 slope -3/4

Normal Form General Equation

5. Angle Between Two Lines and Parallel Perpendicular

Angle θ between two lines slopes m1 m2 tanθ=|(m2-m1)/(1+m1m2)| acute. Example m1=2 m2=0.5 tanθ=0.75 θ≈36.87°.

Parallel m1=m2 Example y=2x+3 and y=2x+5 both slope 2 parallel.

Perpendicular m1×m2=-1 negative reciprocal Example m1=2 m2=-1/2 product -1 perpendicular 90°.

Example Check 3x+4y=7 and 4x-3y=5 perpendicular? m1=-3/4 m2=4/3 product -1 perpendicular true.

Angle θ slopes m1 m2 tanθ=|(m2-m1)/(1+m1m2)| Example m1=2 m2=0.5 tanθ=0.75 θ≈36.87° | Parallel m1=m2 Example y=2x+3 y=2x+5 slope 2 parallel | Perpendicular m1×m2=-1 Example 2 and -1/2 product -1 90° | Example 3x+4y=7 and 4x-3y=5 m1=-3/4 m2=4/3 product -1 perpendicular

Angle Between Two Lines

6. Distance - Point to Line and Between Parallel Lines

Distance point (x0,y0) from line Ax+By+C=0 d=|Ax0+By0+C|/√(A²+B²) absolute shortest perpendicular. Example point (1,2) line 3x+4y-5=0 d=|3+8-5|/5=6/5=1.2.

Distance parallel lines Ax+By+C1=0 and Ax+By+C2=0 same A,B different C d=|C1-C2|/√(A²+B²). Example 3x+4y+5=0 and 3x+4y+10=0 d=|5-10|/5=1.

Distance Formulas

7. Practical Applications - Roads Architecture GPS

Roads Highway design straight sections slope gradient rise over run Positive rising negative falling Civil engineers slope m design safe gradients.

Architecture Buildings roofs walls straight lines intersection angles tanθ ensures perpendicular walls 90°.

GPS navigation Shortest distance point to line formula GPS find nearest road to current location.

Computer graphics Lines drawn screen y=mx+c pixels. Economics Cost vs production linear relationship cost=fixed+variable×quantity straight line slope variable cost.

Example Road (0,0) to (100m,30m rise) slope=0.3=tanθ θ≈16.7° gentle safe.

Roads Highway design straight slope gradient rise over run Positive rising negative falling Civil engineers slope m safe gradients | Architecture Buildings roofs walls straight lines intersection angles tanθ perpendicular 90° | GPS navigation Shortest distance point to line GPS nearest road | Computer graphics y=mx+c pixels Economics cost=fixed+variable×quantity slope variable cost | Example Road (0,0) to (100m,30m) slope 0.3 θ≈16.7° gentle safe

Practical Applications

8. Worked Examples and Formulas Short Tricks

Example 1 Through (2,3) slope 4 y-3=4(x-2) y=4x-5.

Example 2 Through (1,2) and (4,6) (y-2)/4=(x-1)/3 →4x-3y+2=0 slope 4/3.

Example 3 Slope intercepts 3x+4y-12=0 slope -3/4 intercept form x/4+y/3=1 a=4 b=3.

Example 4 Distance (1,2) from 3x+4y-5=0 d=6/5=1.2.

Example 5 Angle between y=2x+3 and y=0.5x+1 slopes 2 and 0.5 tanθ=0.75 θ≈36.87°.

Formulas must remember Slope m=(y2-y1)/(x2-x1)=tanθ Slope-intercept y=mx+c Point-slope y-y1=m(x-x1) Two-point (y-y1)/(y2-y1)=(x-x1)/(x2-x1) Intercept x/a+y/b=1 Normal x cos ω + y sin ω =p General Ax+By+C=0 slope -A/B Angle tanθ=|(m2-m1)/(1+m1m2)| Parallel m1=m2 Perpendicular m1m2=-1 Distance point d=|Ax0+By0+C|/√(A²+B²) Distance parallel |C1-C2|/√(A²+B²).

Example 1 Through (2,3) slope 4 y=4x-5 | Example 2 Through (1,2) (4,6) →4x-3y+2=0 slope 4/3 | Example 3 Slope intercepts 3x+4y-12=0 slope -3/4 x/4+y/3=1 | Example 4 Distance (1,2) from 3x+4y-5=0 d=6/5=1.2 | Example 5 Angle y=2x+3 and y=0.5x+1 tanθ=0.75 θ≈36.87° | Formulas Slope m=(y2-y1)/(x2-x1)=tanθ y=mx+c y-y1=m(x-x1) Two-point Intercept x/a+y/b=1 Normal x cos ω + y sin ω =p General Ax+By+C=0 slope -A/B Angle tanθ=|(m2-m1)/(1+m1m2)| Parallel m1=m2 Perpendicular m1m2=-1 Distance d=|Ax0+By0+C|/√(A²+B²) Parallel |C1-C2|/√(A²+B²)

Worked Examples Formulas