Straight Lines PDF
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Uploaded by LustrousCyclops
Narayana School
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This document provides formulas and equations related to straight lines in coordinate geometry. It covers topics such as slope, distance, angle between lines, and equations of lines. The material is suitable for undergraduate-level mathematics courses.
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# STRAIGHT LINES - slope of line => m = tan θ - slope of x-axis = 0 - slope of y-axis = ∞ - If L<sub>1</sub> || L<sub>2</sub>, m<sub>1</sub> = m<sub>2</sub>. - If L<sub>1</sub> ⊥ L<sub>2</sub>, m<sub>1</sub> * m<sub>2</sub> = -1. - angle between 2 lines => tan θ = (m<sub>2</sub> - m<sub>1</sub>) /...
# STRAIGHT LINES - slope of line => m = tan θ - slope of x-axis = 0 - slope of y-axis = ∞ - If L<sub>1</sub> || L<sub>2</sub>, m<sub>1</sub> = m<sub>2</sub>. - If L<sub>1</sub> ⊥ L<sub>2</sub>, m<sub>1</sub> * m<sub>2</sub> = -1. - angle between 2 lines => tan θ = (m<sub>2</sub> - m<sub>1</sub>) / (1 + m<sub>1</sub> * m<sub>2</sub>) - Slope of line through points => m = (y<sub>2</sub> - y<sub>1</sub>) / (x<sub>2</sub> - x<sub>1</sub>) = (y<sub>2</sub> - y<sub>1</sub>) / (x<sub>2</sub> - x<sub>1</sub>) - slope of line passing through origin => y = mx - Two point form => (y - y<sub>1</sub>) / (x - x<sub>1</sub>) = (y<sub>2</sub> - y<sub>1</sub>) / (x<sub>2</sub> - x<sub>1</sub>) - Slope - intercept form: - x-axis => y = mx - mc - y-axis => y = mx + c - intercept form: x/a + y/b = 1, x/a + y/b = 1 - normal form: bx+ay=ab, xat by =ab, xcos α + ysin α = p. ## ax+by+c = 0 - slope = -a/b - x-intercept = -c/a - y-intercept = -c/b ## EQUATION OF LINES - equation of line parallel to ax+by+c=0 is of the form ax+by+k=0 - equation of line perpendicular to ax+by+c=0 is of the form bx-ay+k=0 ## Symmetric Form - x-= x<sub>1</sub> / cos θ, y-y<sub>1</sub> / sin θ # DISTANCES - perpendicular distance from origin = |c| / √a<sup>2</sup>+b<sup>2</sup>. - perpendicular distance from point = |ax<sub>1</sub> +by<sub>1</sub>+c| / √a<sup>2</sup>+b<sup>2</sup>. - Distance of point with inclination θ = |ax + by + c| / acos θ + bsin θ - distance between parallel lines = |c<sub>1</sub> - c<sub>2</sub>| / √a<sup>2</sup>+b<sup>2</sup> - distance between parallel lines having inclination θ = |c<sub>1</sub> - c<sub>2</sub>| / |acos θ + bsin θ| - equation of line lying midway between parallel lines = ax + by + (c<sub>1</sub> + c<sub>2</sub>) / 2 = 0 # ANGLE BETWEEN LINES θ is an acute angle between lines: a<sub>1</sub>x + b<sub>1</sub>y + c<sub>1</sub> = 0, a<sub>2</sub>x+b<sub>2</sub>y+c<sub>2</sub> = 0, - cos θ = (a<sub>1</sub>a<sub>2</sub> + b<sub>1</sub>b<sub>2</sub>) / √a<sub>1</sub><sup>2</sup>+b<sub>1</sub><sup>2</sup> * √a<sub>2</sub><sup>2</sup>+b<sub>2</sub><sup>2</sup> - tan θ = (a<sub>1</sub>b<sub>2</sub> - a<sub>2</sub>b<sub>1</sub>) / (a<sub>1</sub>a<sub>2</sub> + b<sub>1</sub>b<sub>2</sub>). - sin θ = (a<sub>1</sub>b<sub>2</sub> - a<sub>2</sub>b<sub>1</sub>) / √a<sub>1</sub><sup>2</sup>+b<sub>1</sub><sup>2</sup> * √a<sub>2</sub><sup>2</sup>+b<sub>2</sub><sup>2</sup> - L<sub>1</sub> = ax + by + c<sub>1</sub> = 0 - L<sub>2</sub> = a<sub>2</sub>x+b<sub>2</sub>y+c<sub>2</sub> = 0. - lines are parallel => a<sub>1</sub> / a<sub>2</sub> = b<sub>1</sub> / b<sub>2</sub> - lines are coincident => a<sub>1</sub> / a<sub>2</sub> = b<sub>1</sub> / b<sub>2</sub> = c<sub>1</sub> / c<sub>2</sub> - lines are perpendicular => a<sub>1</sub>a<sub>2</sub>+b<sub>1</sub>b<sub>2</sub>= 0. # FOOT 2 IMAGE - foot => h-x<sub>1</sub> = K-y<sub>1</sub> = -(ax<sub>1</sub>+by<sub>1</sub>+c) / a<sup>2</sup>+b<sup>2</sup> - image => h-x<sub>1</sub> = K-y<sub>1</sub> = -2 (ax<sub>1</sub>+by<sub>1</sub>+c) / a<sup>2</sup>+b<sup>2</sup> - (h,k) = (x<sub>1</sub> + aλ, y<sub>1</sub> + bλ) - λ = -2(ax<sub>1</sub> + by<sub>1</sub> + c) / a<sup>2</sup>+b<sup>2</sup> - image of line ax+by+c=0 w.r.t line lx+my+ n = 0 is (l<sup>2</sup>+m<sup>2</sup>) (ax+by+c) = 2(al+bm) (lx+my+n) # TRIANGLES & QUADRILATERALS - d<sub>1</sub> = distance btw ax+by+c<sub>1</sub>=0; ax + by + c<sub>2</sub>=0. - d<sub>2</sub> = distance btw a<sub>2</sub>x+b<sub>2</sub>y+k<sub>1</sub> = 0; a<sub>2</sub>x+b<sub>2</sub>y+k<sub>2</sub> = 0 - a) square - d<sub>1</sub> = d<sub>2</sub>, a<sub>1</sub>a<sub>2</sub>+b<sub>1</sub>b<sub>2</sub> = 0 - b) rhombus - d<sub>1</sub> = d<sub>2</sub>, a<sub>1</sub>a<sub>2</sub>b+ b<sub>1</sub>b<sub>2</sub> ≠0 - c) rectangle. - d<sub>1</sub> ≠ d<sub>2</sub>, a<sub>1</sub>a<sub>2</sub> + b<sub>1</sub>b<sub>2</sub> = 0 - d) parallelogram, d<sub>1</sub> ≠ d<sub>2</sub>, a<sub>1</sub>a<sub>2</sub>+b<sub>1</sub>b<sub>2</sub> ≠0 - Area of Δ formed by x/a + y/b = 1 => ½ (ab) - Area of Δ formed by ax+by+c=0 => c<sup>2</sup> / 2(ab) - Area of rhombus ax+by+c=0 => 2c<sup>2</sup> / (ab) - p<sub>1</sub> , p<sub>2</sub> are distances btw parallel lines. - θ is angle btw sides - Area of parallelogram = (p<sub>1</sub> p<sub>2</sub>) / sin θ - Area of parallelogram = (c<sub>1</sub>-c<sub>2</sub>)(d<sub>1</sub>-d<sub>2</sub>) / (a<sub>1</sub>b<sub>2</sub> - a<sub>2</sub>b<sub>1</sub>) - ax<sub>1</sub>+ by<sub>1</sub>+c<sub>1</sub> = 0, ax<sub>1</sub>+by<sub>1</sub>+c<sub>2</sub> - ax<sub>2</sub>+by<sub>2</sub>+d<sub>1</sub>=0, ax<sub>2</sub>+by<sub>2</sub>+d<sub>2</sub> - Area of rhombus = ½ * d<sub>1</sub> * d<sub>2</sub>, d<sub>1</sub> & d<sub>2</sub> are diagonals # NO. OF LINES DRAWN: THROUGH A - d = distance from B - a) AB = d => 1 - b) AB >d => 2 - AB = distance btw A & B, c) AB < d => 0 # NO. OF RIGHT ANGLED Δ: IN A CIRCLE - h = height & r = radius - 1) h = r => 2 - 2) h<r => 1 - 3) h > r => 0 # ANGULAR BISECTOR - a<sub>1</sub>x + b<sub>1</sub>y + c<sub>1</sub>=0, a<sub>2</sub>x + b<sub>2</sub>y + c<sub>2</sub> = 0 - (a<sub>1</sub>x+b<sub>1</sub>y+c<sub>1</sub>) / √a<sub>1</sub><sup>2</sup>+b<sub>1</sub><sup>2</sup> = (a<sub>2</sub>x+b<sub>2</sub>y+c<sub>2</sub>) / √a<sub>2</sub><sup>2</sup>+b<sub>2</sub><sup>2</sup> - c<sub>1</sub>c<sub>2</sub> (a<sub>1</sub>a<sub>2</sub>+b<sub>1</sub>b<sub>2</sub>) < 0 origin in acute - c<sub>1</sub>c<sub>2</sub> (a<sub>1</sub>a<sub>2</sub>+b<sub>1</sub>b<sub>2</sub>) > 0 origin in obtuse # NO. OF CIRCLES TOUCHING ALL THREE LINES: - a) no circle if lines are parallel - b) one circle if lines are concurrent - c) two circles if two lines are parallel - d) four circles if lines are not concurrent & not parallel. # PAIR OF STRAIGHT LINES - General form => ax<sup>2</sup> + 2hxy + by<sup>2</sup> - Seperate equations: - ax+(h+ √h<sup>2</sup>-ab) y = 0 - ax + (h-√h<sup>2</sup>-ab) y = 0 # NATURE OF LINES: - h<sup>2</sup>-ab > 0 => real & distinct - h<sup>2</sup>- ab = 0 => coincident lines - h<sup>2</sup>-ab < 0 => imaginary lines. # SLOPE OF LINES - y<sup>2</sup> - (m<sub>1</sub>+ m<sub>2</sub>) xy + m<sub>1</sub>m<sub>2</sub>x<sup>2</sup> - m<sub>1</sub> + m<sub>2</sub> = -2h - |m<sub>1</sub> - m<sub>2</sub>| = 2√h<sup>2</sup>-ab / |b| - Slopes are in ratio m<sub>1</sub>:m<sub>2</sub> => (m+n)<sup>2</sup> ab = 4 h<sup>2</sup> mn - Slopes are m<sub>1</sub>:m<sub>2</sub><sup>n</sup>; (ab)<sup>1/n+1</sup> + (ab)<sup>1/n+1</sup> + 2h = 0 # ANGLE BETWEEN PAIR OF LINES - θ is the acute angle - sine = 2√h<sup>2</sup>-ab / (a+b) - cos θ = √(a-b)<sup>2</sup> + 4h<sup>2</sup> / (a+b) - tan θ = 2√h<sup>2</sup>-ab / (a+b) - θ = 0°, x<sup>2</sup>-ab=0, parallel - θ = 90°, a+b=0, perpendicular # TYPES OF TRIANGLES: - ax<sup>2</sup> + 2hxy + by<sup>2</sup>, lx + my + n = 0. - a) equilateral => ax<sup>2</sup>+2hxy+by<sup>2</sup> = (lx+my)<sup>2 </sup>- 3 (mx - ly)<sup>2 </sup>. - b) isosceles => h (l<sup>2</sup>-m<sup>2</sup>) = (a-b) lm - c) right angeled => a+b=0 (or) a<sup>2</sup>l<sup>2</sup> + 2hlm + b<sup>2</sup>m<sup>2</sup> = 0 - (lx+my)<sup>2</sup> - tan<sup>2</sup>α (mx-ly)<sup>2 </sup>= 0 - a) tan<sup>2</sup> α = 3, equilateral - b) tan<sup>2</sup>α = 1, right angled isoceles - c) tan<sup>2</sup>α<1, isosceles & obtuse angled. # AREA OF TRIANGLE - ax<sup>2</sup> + 2hxy + by<sup>2</sup>, lx + my + n = 0 - Area of Δ = h<sup>2</sup> √h<sup>2</sup>-ab / (l<sup>2</sup>+m<sup>2</sup>)<sup>2</sup> - (lx+my)<sup>2</sup> - tan<sup>2</sup>α (mx-ly)<sup>2 </sup>= 0 - Area of Δ = n<sup>2</sup> / (l<sup>2</sup>+m<sup>2</sup>) - Area of equilateral Δ = n<sup>2</sup> /√3 (l<sup>2</sup>+m<sup>2</sup>) = p<sup>2</sup> / √3. - p = perpendicular d (from 0,0) = n / √l<sup>2</sup>+m<sup>2</sup> - (lx+my)<sup>2 </sup> - k(mx - ly)<sup>2</sup> = 0, (lx+my)<sup>2 </sup>- 3(mx - ly)<sup>2 </sup>= 0 # CENTRES RELATED TO TRIANGLES: - (α, β) is centroid, ax<sup>2</sup> + 2hxy + by<sup>2</sup>, lx+my+n=0 - α = bl-hm / 3(am<sup>2</sup>-2hlm+bl<sup>2</sup>) - β = am-hl / 3(am<sup>2</sup>-2hlm+bl<sup>2</sup>) - (K, K<sub>m</sub>) is orthocentre, ax<sup>2</sup> + 2 hay tby<sup>2</sup> = 0 , lx+my+n=0 - K = -n (a+b) / am<sup>2</sup> - 2hml + bl<sup>2</sup>. - (l,m) is orthocentre, ax<sup>2</sup> + 2hxy + by<sup>2</sup> = 0. - Eq of 3rd side = (a+b)(lx+my) = am<sup>2</sup> - 2hlm+bl<sup>2</sup>. - S = ax<sup>2</sup> + 2hxy+by<sup>2</sup> = 0, (x<sub>1</sub>,y<sub>1</sub>) is midpoint of 3rd side - ax<sub>1</sub> + h(xy<sub>1</sub>+x<sub>1</sub>y) + by<sub>1</sub> = ax<sub>1</sub><sup>2</sup> + 2hx<sub>1</sub>y<sub>1</sub> + by<sub>1</sub><sup>2</sup>. - (x,y) is centroid, S = ax<sup>2</sup> + 2 hay + by<sup>2</sup> = 0 - S<sub>1</sub> = 1/3 S<sub>11</sub> - ax<sub>1</sub> + hexy<sub>1</sub> + yx<sub>1</sub>) + by<sub>1</sub> = 1/3 (ax<sub>1</sub><sup>2</sup> + 2hxiy<sub>1</sub>+by<sub>1</sub><sup>2</sup>) # INTERCEPT OF PAIR OF LINES - ax<sup>2</sup> + 2hxy + by<sup>2</sup>=0, lx+my+n=0 - 2 √(h<sup>2</sup>-ab)(l<sup>2</sup>+m<sup>2</sup>)/am<sup>2</sup>-2him + bl<sup>2</sup>. # - If aix<sup>2</sup> + 2nixy + biy<sup>2</sup> = 0, a<sub>2</sub>x<sup>2</sup>+2hz xy + b<sub>2</sub>y<sup>2</sup> = 0, one line in common, (aibz-azbi)<sup>2</sup> + 4 (hiaz-hza1) (hibz-habi) = 0 - If one of the lines are perpendicular to another one of the lines, (aiaz-bib₂)<sup>2</sup> + 4 (hiaz-habı) (hibz-h2a1) = 0 - one line in common, other lines are perpendicular, then hia2 b2 - hza1b1 = 1 / √aiaz bibz - ax<sup>2</sup> + 2nxy+by<sup>2</sup>, (x<sub>1</sub>,y<sub>1</sub>) - (i) parallel => a (x-x<sub>1</sub>)<sup>2</sup> + 2h (x-x<sub>1</sub>)(y-y<sub>1</sub>)+b(y-y<sub>1</sub>)<sup>2</sup>=0 - (ii) perpendicular => b (x-x<sub>1</sub>)<sup>2</sup> + 2h(x-x<sub>1</sub>)(y-y<sub>1</sub>) + a(y-y<sub>1</sub>)<sup>2</sup> = 0 - ax<sup>2</sup> + 2hay + by<sup>2</sup> = 0 , lx+my+n=0 - (i) parallel => am<sup>2</sup> - 2hlm + bl<sup>2</sup> = 0 - (ii) perpendicular => al<sup>2</sup> + anim + bm<sup>2</sup>=0 # ANGULAR BISECTORS: - Equation of pair of bisectors = h (x<sup>2</sup>-y<sup>2</sup>) = (a-b)xy - Equation to the bisectors of coordinate axis => x<sup>2</sup>- y<sup>2</sup> = 0 - One of the line in ax<sup>2</sup> + 2hxy + by<sup>2</sup> = 0 bisects the angle between coordinate axes = (a+b)<sup>2</sup> = 4h<sup>2</sup>. - angle between pair of bisectors is 90° (or) √2.
