Pair of Straight Lines PDF

MATRIX ACADEMY, PAROLA By Prof. Hemant Pawar Chapter 4 : Pair of Straight Lines  Distance formulae – If A = ( x1 ,y1 ) and B = (x2 , y2) are two point...

MATRIX ACADEMY, PAROLA By Prof. Hemant Pawar Chapter 4 : Pair of Straight Lines  Distance formulae – If A = ( x1 ,y1 ) and B = (x2 , y2) are two points then the distance between A &B is : d(AB) = (𝑥𝑥2 − 𝑥𝑥1 )2 + (𝑦𝑦2 − 𝑦𝑦1 )2  Inclination of a line – Y The smallest angle made by a line with the positive X - axis in the anticlockwise sense is called as inclination of line. Inclination of line is denoted by " 𝜃𝜃 " ɵ where , 0° ≤ 𝜃𝜃 ≤ 180° x  Slope of a line – If the inclination of a line is 𝜃𝜃 then tan𝜃𝜃 is called as slope of a line. Slope of line is denoted by ' m ' m = tan𝜃𝜃 is slope. If A = ( x1 ,y1 ) and B = (x2 , y2) are the two points given on the line the slope is 𝑦𝑦2−𝑦𝑦1 𝑦𝑦1−𝑦𝑦2 m= OR 𝑥𝑥2−𝑥𝑥1 𝑥𝑥1−𝑥𝑥2 Hence , 𝑦𝑦2−𝑦𝑦1 𝑦𝑦1−𝑦𝑦2 slope m = OR m= = tan𝜃𝜃 𝑥𝑥2−𝑥𝑥1 𝑥𝑥1−𝑥𝑥2 1 Prof. Hemant Pawar Sir`s Mathematics Notes Mob. No. - 9595209515  MATRIX ACADEMY, PAROLA By Prof. Hemant Pawar  Equation of a line – 1 ) General equation of lines – General equation of line is : ax + by + c = 0 Where a ,b, c are constants And x , y , z are variables. 2 ) Slope intercept form –  If a line has slope m and y – intercept c then it's y equation in slope-point form is : m = tanɵ ɵ y = mx + c x 3 ) Slope - point form – If a line passing through the point ( x ,y ) and having a slope m then it's equation in slope –point y form is :- m = tanɵ ɵ y - y₁ = m ( x – x1 ) x 4] Two - point form – Y If a line passing through two points ( x1 ,y1 ) and (x2 , y2) then it's equation in two point form is :- ( x1 ,y1 ) (x2 , y2) 𝑦𝑦−𝑦𝑦1 𝑦𝑦−𝑦𝑦2 = x 𝑥𝑥−𝑥𝑥1 𝑥𝑥−𝑥𝑥2 2 Prof. Hemant Pawar Sir`s Mathematics Notes Mob. No. - 9595209515  MATRIX ACADEMY, PAROLA By Prof. Hemant Pawar 5] Double intercept forms:- y If a line having a & b intercepts on X- axis and Y B(0,b) – axis then it's equation in double intercept form is :- A(a,0) 𝑥𝑥 𝑦𝑦 x + =1 𝑎𝑎 𝑏𝑏 6 ) Normal form – y The equation of a line whose perpendicular distance from the origine is ' p ' and ɑ is inclination of its normal is given by :- Normal P x cosɑ + y sinɑ = p ɑ x  Note :- 1) Equation of line passing through origin is y = mx +c and ax + by + c = 0 𝒂𝒂 2) Slope of line having equation ax + by + c = 0 is m = 𝒃𝒃 −𝒄𝒄 ⇒ X - intercept is 𝒂𝒂 −𝒄𝒄 ⇒ Y- intercept is 𝒃𝒃 3) Equation of X – axis is :- y = 0 4) Equation of X - axis is : - x = 0 5) Equation of line parallel with x - axis is: y = b 6) Equation of line parallel with y– axis is : x = a 3 Prof. Hemant Pawar Sir`s Mathematics Notes Mob. No. - 9595209515  MATRIX ACADEMY, PAROLA By Prof. Hemant Pawar  Angle between two straight lines :- If m1 & m₂ are slopes of two straight lines then the acute angle between these two lines is : | m1 – m2 | tan ɵ = If 𝑚𝑚1. 𝑚𝑚2 ≠ −1 |1 + 𝑚𝑚1 𝑚𝑚2 |  Note :- Two lines having slopes m1 & m2 are i) Parallel if m₁ = m2 and ii) Perpendicular if m1.. m 2 = -1  Perpendicular distance from point to line :- The length of perpendicular from a point ( x , y ) to the line ax + by + c = 0 is :- |ax1 + by1 + c| d = | √𝑎𝑎2 + 𝑏𝑏 2 |  Combined equation of pair of lines :- An equation which represents two lines is called the combined equation of those two lines. combined equation is also called as Joint equation. 4 Prof. Hemant Pawar Sir`s Mathematics Notes Mob. No. - 9595209515