Tiêu đề Unit-2,Straight line.pdf ✅

DEPARTMENT OF TECHNICAL EDUCATION,BENGALORE GOVERNMENT POLYTECHNIC, RAICHUR 2020-21 ENGINEERING MATHEMATICS UNIT-II, STRAIGHT LINE STUDY MATERIALS Prepared by: Ramachandra Sutar N E A R G O V T I T I C O L L E G E A M A R K H E D L A Y O U T R A I C H U R - 5 8 4 1 0 3
Department of Science _Govt Polytechnic, Raichur_Ramachandra Sutar 2 UNIT-2, STRAIGHT LINES 2.1 Syllabus: • Slope of the straight line(provided with inclination and two points on the line as well) and problems • Intercepts of a straight line and problems • Intercept form of a straight line and problems • Slope-intercept form of a straight line and problems • Slope-point form of the straight line and problems • Two-point form of a straight line and problems • General form of a straight line. Problems on finding slope and intercepts. • Angle between two straight lines and conditions for the lines to be parallel and perpendicular and problems • Equation of a line parallel to the given line and Problems • Equation of a line perpendicular to the given line. problems 2.2 Recapitulation: Analytical geometry deals with the study of geometry using the principles of algebra. It is also called as Algebraic Geometry or Co-ordinate Geometry. French philosopher and Mathematician Rene Descartes (1596-1665) introduced the concept of representing a point in a plane by an ordered pair of real numbers. This pair of numbers is called the Cartesian coordinates of the point named after Rene Descartes. The plane in which the points are represented is called the Cartesian plane. Straight line is the locus of a moving point P (x,y), which moves in such a condition that P is always collinear with the given two fixed points. Also, we know from the concept of co linearity that, three points are collinear if they lie on a same straight line or the area of triangle from by these points is zero. A line can be identified with a linear equation in x and y. Recall the following formulae. 1 Two straight lines which are mutually perpendicular to each other are called co-ordinate axes. The horizontal line is called x-axis, vertical line is called y-axis. The point of intersection of axis is called the origin. The coordinates of origin are (0,0) 2 Any point on x-axis is given by (x,0) and any point on y-axis is given by (0,y). The point in first quadrant is p(x,y) 3 Distance formula: The distance between two points A(x1,y1) and B(X2,y2) is given by AB= ( ) ( ) 2 2 1 2 2 1 x − x + y − y Or AB= ( ) ( ) 2 1 2 2 1 2 x − x + y − y 4 Section formula :The point P(x,y) which divides the line joining the points A(x1,y1) & B(x2,y2) internally in the ratio m:n is given by P(x, y) =       + + + + m n my ny m n mx nx 2 1 2 1 , X Y O (x,0) (0,y) P(x,y) A(x1,y1) B(x2,y2) A(x1,y1) P(x, y) B(x2,y2) Q(x, y)
Department of Science _Govt Polytechnic, Raichur_Ramachandra Sutar 3 The point P(x ,y) which divides the line joining points A(x1,y1) & B(x2,y2) externally in the ratio m:n is given by Q(x, y) =       − − − − m n my ny m n mx nx 2 1 2 1 , 5 Mid-Point formula: If the point P(x , y) which divides the line joining points A(x1,y1) & B(x2,y2) equally I,e in the ratio m:m then the point P(x , y) is called mid-point of the line joining points A & B and is given by P(x, y) =       + + 2 , 2 1 2 1 2 x x y y 6 Centroid of a triangle: The centroid of a triangle ABC whose vertices are A(x1,y1) , B(x2,y2) & C(x3,y3) is given by G(x, y) =       + + + + 3 , 3 1 2 3 1 2 3 x x x y y y 7 Locus of a point: It is the path traced by a point which according to the given conditions. Ex: If a point moves so that its distance from a fixed point is always constant. The path traced by the point is called is circle and the fixed point is called centre, and the distance is called radius of a circle. 2.3 Slope of a straight line: If a Straight line makes an angle  (0    ) with the positive direction of x-axis then the angle  is called the inclination of the line, and the real number tan denoted by m is called the slope of the straight line. I,e m=tan  The slope of line joining points A(x1, y1) & B(x2,y2) is given by m= 2 1 2 1 x x y y − − Note: • If a line is parallel to x-axis then  =0 and m=tan00=0.  Slope of any line parallel to x-axis is 0 • In particular, the slope of x-axis is 0. • If a line is perpendicular to x-axis, then  =900 and m=tan900=  .  Slope of any line perpendicular to x-axis (parallel to y-axis) is  . • In particular the slope of y-axis is  . • Two line are said to be parallel if their slopes are equal.[m1=m2] • Two lines are said to be perpendicular if product of their slopes are equal to -1.[m1 m2=-1] Where m1 is the slope of first line and m2 is the slope of second line] 2.3.1 Problems on slope of line when inclination (angle) is given: 1. Find slope of a straight line which makes angle 300 with the positive direction of x-axis. Given: Inclination  = 300 W.K.T Slope of straight line is A B C G X-axis Y-axis
Department of Science _Govt Polytechnic, Raichur_Ramachandra Sutar 4 3 1 tan30 tan 0 = = = m m m  2. Find slope of a straight line which makes angle 600 with the positive direction of x-axis. Given: Inclination  = 600 W.K.T Slope of straight line is 3 tan 60 tan 0 = = = m m m  3. Find slope of a straight line which makes angle 450 with the positive direction of x-axis. Given: Inclination  = 450 W.K.T Slope of straight line is 1 tan 45 tan 0 = = = m m m  4. Find slope of a straight line which makes angle 900 with the positive direction of x-axis. Given: Inclination  = 900 W.K.T Slope of straight line is =  = = m m m 0 tan90 tan 5. Find slope of a straight line which makes angle 1350 with the positive direction of x-axis. Given: Inclination  = 1350 W.K.T Slope of straight line is 1 tan 45 tan(180 45 ) tan135 tan 0 0 0 = − = − = − = = m m m m m  6. Find slope of a straight line which makes angle 150 0 with the positive direction of x-axis. Given: Inclination  = 150 0 W.K.T Slope of straight line is 3 1 tan30 tan(180 30 ) tan150 tan 0 0 0 = − = − = − = = m m m m m  2.3.2 Problems on slope of straight line when its line joining points are given: 1. Find the slope of line joining points (3,-2) and (7, 5) Given: line joining points (3,-2) and (7, 5) W.K.T Slope of straight line is