Straight Line Notes

Page # 1 STRAIGHT LINE STRAIGHT LINE 1. EQUATION OF STRAIGHT LINE A relation between x and y which is satisfied by co-ordinates of every point lying on a line is called the equation of Straight Line. Every linear equation in two variable x and y always represents a straight line. eg. 3x + 4y = 5, – 4x + 9y = 3 etc. General form of straight line is given by ax + by + c = 0. 2. EQUATION OF STRAIGHT LINE PARALLEL TO AXES (i) Equation of x axis  y = 0. Equation a line parallel to x axis (or perpendicular to y axis) at a distance ‘a’ from it y = a. (ii) Equation of y axis  x = 0. Equation of a line parallel to y axis (or perpendicular to x axis) at a distance ‘a’ from it  x = a. eg. Equation of a line which is parallel to x-axis and at a distance of 4 units in the negative direction is y = – 4. 3. SLOPE OF A LINE If  is the angle made by a line with the positive direction of x axis in anticlockwise sense, then the value of tan is called the Slope (also called gradient) of the line and is denoted by m or slope = tan  eg. A line which is making an angle of 45º with the x-axis then its slope is m = tan 45º = 1. Note : (i) Slope of x axis or a line parallel to x axis is tan 0º = 0. (ii) Slope of y axis or a line parallel to y axis is tan 90º =  . (iii) The slope of a line joining two points y 2  y1 (x1, y1) and (x2, y2) is given by m = x  x . 2 1 eg. Slope of a line joining two points (3, 5) and 95 4  = 1. (7, 9) is = 7 3 4 4. DIFFERENT FORMS OF THE EQUATION OF STRAIGHT LINE 4.1 Slope - Intercept Form : The equation of a line with slope m and making an intercept c on y–axis is y = mx + c. If the line passes through the origin, then c = 0. Thus the 1 equation of a line with slope m and passing through the origin y = mx. 4.2 Slope Point Form : The equation of a line with slope m and passing through a point (x1, y1) is y – y1 = m(x – x1) 4.3 Two Point Form : The equation of a line passing through two given points (x1, y1) and (x2, y2) is – y 2  y1 y – y1 = x  x (x – x1) 2 1 4.4 Intercept Form : The equation of a line which makes intercept a and b on the x axis and y axis respectively is x y  =1. Here, the length of intercept between a b the co-ordinates axis = a 2  b 2 B Y (0, b) b (a, 0) O a A X 1 1 OA. OB = a.b. 2 2 4.5 Normal (Perpendicular) Form of a Line : If p is the length of perpendicular on a line from the origin and  is the inclination of perpendicular with x- axis then equation on this line is xcos + ysin = p Area of OAB = 4.6 Parametric Form (Distance Form) : If  be the angle made by a straight line with x-axis which is passing through the point (x1, y1) and r be the distance of any point (x, y) on the line from the point (x1, y1 ) then its equation. y  y1 x  x1 = =r cos  sin  5. REDUCTION OF GENERAL FORM OF EQUATIONS INTO STANDARD FORMS General Form of equation ax + by + c = 0 then its - (i) Slope Intercept Form is y = – slope m = – a c , Intercept C = b b a c x – , here b b Page # 2 STRAIGHT LINE (ii) Intercept Form is x y + = 1, here c/a c/b x intercept is = – c/a, y intercept is = – c/b (iii) Normal Form is To change the general form of a line into normal form, first take c to right hand side and make it positive, then divide the whole equation by by ax c   , a 2  b 2 like – a 2  b 2 2 2 2 a b a  b2 here cos = – and p = a a 2  b2 , sin = – c b a 2  b2 2 a  b2 Note : Lines a1x + b1y + c1 = 0 and a 1 b1 a2x + b2y + c2 = 0 are parallel  a  b 2 2 7.2 Perpendicular Lines : Two lines are perpendicular, then angle between them is 90º m1  m 2  1  m m = tan90º =   m1m2 = –1 1 2 Note : Lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are perpendicular then a1a2 + b1b2 = 0 7.3 Coincident Lines : Two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 are coincident only and only if a 1 b1 c1  a  b  c 2 2 2 6. POSITION OF A POINT RELATIVE TO A LINE (i) The point (x1, y1) lies on the line ax + by + c = 0 if, ax1+ by1 + c = 0 (ii) If P(x1, y1) and Q(x2, y2) do not lie on the line ax + by + c = 0 then they are on the same side of the line, if ax1+by1+ c and ax2 + by2 +c are of the same sign and they lie on the opposite sides of line if ax1 + by1 + c and ax2 + by2 + c are of the opposite sign. (iii) (x1, y1) is on origin or non origin sides of the line ax + by + c = 0 if ax1 + by1 + c = 0 and c are of the same or opposite signs. 8. EQUATION OF PARALLEL & PERPENDICULAR LINES (i) Equation of a line which is parallel to ax + by + c = 0 is ax + by + k = 0 (ii) Equation of a line which is perpendicular to ax + by + c = 0 is bx – ay + k = 0 The value of k in both cases is obtained with the help of additional information given in the problem. 7. ANGLE BETWEEN TWO STRAIGHT LINES The an gle betw een two strai ght l in es y = m1x + c1 and y = m2x + c2 is given by 9. EQUATION OF STRAIGHT LINES THROUGH (X1,Y1)MAKING AN ANGLE  WITH y = mx + c m1  m 2 tan  = 1  m m 1 2 Y P(x1y1)  Note : (i) If any one line is parallel to y axis then the angle 1 between two straight line is given by tan =  m Where m is the slope of other straight line (ii) If the equation of lines are a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 then above formula would a1b 2  b1a 2 be tan  = a a  b b 1 2 1 2 (iii) Here two angles between two lines, but generally we consider the acute angle as the angle between them, so in all the above formula we take only positive value of tan. 7.1 Parallel Lines : Two lines are parallel, then angle between them is 0 m1  m 2  1  m m = tan0º = 0 1 2  m1 = m2  O x y = mx + c m  tan  y – y1 = (x – x1) 1  m tan  10. LENGTH OF PERPENDICULAR The length P of the perpendicular from the point (x1, y1) on the line ax + by + c = 0 is given by P= ax1  by1  c a 2  b2 Note : (i) Length of perpendicular from origin on the line ax + by + c = 0 is c a 2  b 2 (ii) Length of perpendicular from the point (x1, y1) on the line x cos  + y sin  = p is x1cos  + y1 sin  = p