CHAPTER 10 Straight lines Learning Objectives (i) Slope (m) of a non-vertical line passing through the points (x 1 ) is given by (ii) If a line makes an angle α with the positive direction of x-axis, then the slope of the line is given by. (iii) Slope of horizontal line is zero and slope of vertical line is undefined. (iv) An acute angle (say θ) between lines L 1 and L 2 with slopes m 1 and m 2 is given by (v) Two lines are parallel if and only if their slopes are equal. (vi) Two lines are perpendicular if and only if product of their slopes is 1. (vii) Three points A, B and C are collinear, if and only if slope of AB = slope of BC. (viii) Equation of the horizontal line having distance a from the x-axis is either y = a or y = a. (ix) Equation of the vertical line having distance b from the y-axis is either x = b or x = b. (x) The point (x, y) lies on the line with slope m and through the fixed point (x 0, y 0 ), if and only if its coordinates satisfy the equation y y 0 = m (x x 0 ). (xi) Equation of the line passing through the points (x 1 ) is given by. (xii) The point (x, y) on the line with slope m and y-intercept c lies on the line if and only if y = mx + c. (xiii) If a line with slope m makes x-intercept d. Then equation of the line is y = m (x d). (xiv) Equation of a line making intercepts a and b on the x-and y-axis, respectively, is. (xv) The equation of the line having normal distance from origin p and angle between normal and the positive x-axis ω is given by x cos ω + y sin ω = p. (xvi) Any equation of the form Ax + By + C = 0, with A and B are not zero, simultaneously, is called the general linear equation or general equation of a line. (xvii) The perpendicular distance (d) of a line Ax + By + C = 0 from a point (x 1 ) is given by. (xviii) Distance between the parallel lines Ax + By + C 1 = 0 and Ax + By + C 2 = 0, is given by. INTRODUCTION Analytic geometry, is done by representing points in the plane by ordered pairs of real numbers, called Cartesian co-ordinates and representing lines and curved by algebraic equations. In analytic geometry because of use of co-ordinate, it is called coordinate geometry. BASIC CONCEPTS OF CO-ORDINATES GEOMETRY : Cartesian Co-ordinates Let XOX and YOY be two perpendicular straight lines drawn through point O in the plane of the paper. Then X Axis : The line XOX is called X axis Y Axis: The line YOY is called Y axis Co-ordinate axes : X axis and Y axis together are called axis of co-ordinates or axis of reference Origin : The point O is called the origin of co-ordinates. Cartesian Co-ordinates : The ordered pair of perpendicular distance from both axis of a point P lying in the plane is called Cartesian co-ordinates of P. 1
If the cartesian co-ordinates of a point P are (x, y) then x is called abscissa or x co-ordinate of P and y is called the ordinate or y co-ordinate of point P. (i) Co-ordinates of the origin is (0, 0) (ii) y co-ordinate on x-axis is zero. (iii) x co-ordinate on y axis is zero. Polar Co-ordinates Let OX be any fixed line which is usually called the initial line and O be a fixed point on it. If distance of any point P from the pole O is r and XOP = θ, then (r, θ) are called the polar co-ordinates of a point P. If (x, y) are the Cartesian co-ordinates of a point P, then x = rcosθ ; y = rsinθ and r =, θ = Distance Formula The distance between two points P(x 1 ) and Q(x 2 ) is given by PQ = (i) Distance of point P(x,y) from the origin = (ii) Distance between two polar co-ordinates A (r 1, θ 1 ) and B (r 2, θ 2 ) is given by AB = Co-ordinate of some particular points : Let A(x 1,y 1 ), B(x 2,y 2 ) and C(x 3,y 3 ) are vertices of any triangle ABC, then 1. Centroid : The centroid is the point of intersection of the medians (Line joining the mid point of sides and opposite vertices) Centroid divides the median in the ratio of 2 : 1. Co-ordinates of centroid G 2. Incentre : The incentre is the point of intersection of internal bisector of the angle. Also it is a centre of circle touching all the sides of a triangle. Co-ordinates of incentre I where a, b, c are the sides of triangle ABC (i) Angle bisector divides the opposite sides in the ratio of remaining sides Ex. (ii) Incentre divides the angle bisectors in the ratio ( b + c) : a, (c + a) : b and (a + b) : c 3. Excentre : Co-ordinate of excentre opposite to A is given by I 1 and similarly for excentres (I 2 & I 3 ) opposite to B and C are given by I 2 I 3 4. Circumcentre : The point of concurrency of the perpendicular bisectors of the sides of a triangle is called circumcentre of the triangle. The coordinates of the circumcentre are given by 2
O 5. Orthocentre : It is the point of intersection of perpendicular drawn from vertices on opposite sides (called altitudes) of a triangle and can be obtained by solving the equation of any two altitudes. If a triangle is right angled triangle, then ortho centre is the point where right angle is formed. (i) If the triangle is equilateral, the centroid, incentre, orthocentre, circumcentre, coincides (ii) Ortho centre, centroid and circumcentre are always colinear and centroid divides the line joining orthocentre and circumcentre in the ratio 2 : 1. (iii) In an isosceles triangle centroid, orthocentre, incentre, circumcentre lies on the same line. Image of an object in a mirror : When an object is placed in front of a plane mirror, then its image is formed at the same distance behind the mirror as the distance of the object from the mirror. Reflection : The transformation R 1 which maps a point P to its image P in a given line (or point), is called a reflection in. Thus, R (P) = P'. We shall represent : (i) Reflection in x-axis by R x, (ii) Reflection in y-axis by R y, (iii) Reflection in the origin by R o, Reflection in x-axis : Let P (x, y) be a point in a plane. Draw PM OX, meeting it at M. Produce PM to P' such that MP = MP'. Then, P' is the image of P when reflected in x-axis. Clearly, the co-ordinates of P' are P' (x, y) P (x, y) when reflected in x-axis, have the image P (x, y) R x (x, y) = (x, y) Reflection in y-axis : Let P (x, y) be a point in a plane. Draw PN OY, meeting it at N. Produce PN to P' such that NP = NP'. Then, P' is the image of P when reflected in y-axis. Clearly, the co-ordinates of P are P (x, y) P (x, y) when reflected in x-axis, have the image P' ( x, y) R y (x, y) = ( x, y) Reflection in the origin : Let P (x, y) be a point in a plane. Join PO and produce it to P such that OP = OP. Then, P is the image of P when reflected in the origin. Clearly, the co-ordinate of P are P ( x, y) P (x, y) when reflect in the origin, has the image P' ( x, y) R 0 (x, y) = ( x, y) Section Formula : (a) If P(x, y) divides the line joining A(x 1 ) & B(x 2 ) in the ratio m : n, then (i) Internal division: (ii) External division: 3
(b) The coordinates of the mid-point of the line-segment joining (x 1 ) are Area of a triangle : Let (x 1 ), (x 2 ) and (x 3, y 3 ) respectively be the coordinates of the vertices A, B, C of a triangle ABC. Then the area of triangle ABC, is [x 1 (y 2 y 3 )+ x 2 (y 3 y 1 ) + x 3 (y 1 y 2 )]...(1) =...(2) In case of polygon with vertices (x 1 ), (x 2 ),... (x n, y n ) in order, then area of polygon is given by (x 1 y 2 y 1 x 2 ) + (x 2 y 3 y 2 x 3 ) +...+ (x n 1 y n y n 1 x n) ) + (x n y 1 y n x 1 ) Collinearity of three given points : Three given points A, B, C are collinear if any one of the following conditions is satisfied. (i) Area of triangle ABC is zero. (ii) Slope of AB = slope of BC = slope of AC. (iii) AC = AB + BC. (iv) Find the equation of line passing through 2 given points, if the third point satisfies the given equation of the line, then three points are collinear. Locus : When a point moves in a plane under certain geometrical conditions, the point traces out a path. This path of the moving point is called its locus. The equation to a locus is the relation which exists between the coordinates of any point on the path, and which holds for no other point except those lying on the path. In other words equation to a curve (or locus) is merely the equation connecting the x and the y coordinates of every point on the curve. Procedure for finding the equation of the locus of a point : (i) If we are finding the equation of the locus of a point P, assign coordinates (h, k) or (x 1 ) to P. (ii) Express the given conditions in terms of the known quantities to facilitate calculations. We sometimes include some unknown quantities known as parameters. (iii) Eliminate the parameter. So that the eliminant contains only h, k and known quantities. If h and k coordinates of the moving point are obtained in terms of a third variable t called the parameter, eliminate t to obtain the relation in h and k and simplify this relation. (iv) Replace h by x, and k by y, in the eliminant. The resulting equation would be the equation of the locus 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. Ex. 4x + 9y = 40, 10x + 6y = 8 etc. General form of straight line is given by ax + by + c = 0. 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 Ex. Equation of a line which is parallel x axis and at a distance of 4 units in the negative direction is y = 4 Slope of a line : The slope of a line is equal to the tangent of the angle which it makes with the positive side of x-axis and it is generally denoted by m. Thus if a line makes an angle θ with x-axis then its slope = m = tanθ 4 The slope of a line joining two points (x 1 ) is given by m = (i) < m (ii) Slope of x axis or a line parallel to x axis is tan 0 = 0 (iii) Slope of y axis or a line parallel to y axis is tan 90 =.
Different forms of the equation of straight line : General form : ax + by + c = 0 where a, b, c are any real numbers and a, b are not zero both at a time. Particular case ; in ax + by + c = 0 a = 0 by + c = 0 which is a line parallel to x- axis b = 0 ax + c = 0 which is a line parallel to y- axis c = 0 ax + by = 0 which is a line passing through origin. Slope form : y = mx + c Where m is the slope of the line and c is the length of the intercept made by it on y-axis for general form ax + by + c = 0 Slope= Slope Point Form : The equation of a line with slope m and passing through a point (x 1 ) is y y 1 = m (x x 1 ). Two Point Form : The equation of a line passing through two given points (x 1 ) is y y 1 = (x x 1 ) Intercept Form : The equation of a line which makes intercept a and b on the x-axis and y-axis respectively is. Here, the length of intercept between the co-ordinates axis = Area of OAB = OA. OB = a.b. of P. Normal form : x cosα + y sinα = p, where α, is the angle which the perpendicular to the line makes with the axis of x and p is the length of the perpendicular from the origin to the line. 0 α < 2π and p is always positive. Parametric form : In figure given below let BAP be a straight line through a given point A (x 1 ), the angle of slope being θ. The positive direction of the line is in the sense BAP. (Direction of increasing ordinate is called the positive direction of the line). For the points P (x,y) and Q (X, Y) Shown in the figure AP is regarded as a positive vector and AQ as a negative vector, as indicated by the arrows. From the general definitions of cosθ and sinθ we have cosθ =, sinθ = or x x 1 = AP cosθ, y y 1 = AP sinθ. 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 =, here slope m =, Intercept C = (ii) Intercept Form is + = 1, here x intercept is = c/a, y intercept is = c/b 5
(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 like., here cosα =, sinα = and p = Position of a given point relative to a given line : The fig. Shows a point P(x 1 ) lying above a given line. If an ordinate is dropped from P to meet the line L at N, then the x coordinate of N will be x 1. Putting x = x 1 in the equation ax + by + c = 0 gives y coordinate of N = If P(x 1 ) lies above the line, then we have y 1 > i.e. y 1 + > 0 i.e. > 0 i.e. > 0...(1) Hence, if P(x 1 ) satisfies equation (1), it would mean that P lies above the line ax + by + c = 0, and if < 0, it would mean that P lies below the line ax + by + c = 0. If (ax 1 + by 1 + c) and (ax 2 + by 2 + c) have same signs, it implies that (x 1 ) both lie on the same side of the line ax + by + c = 0. If the quantities ax 1 + by 1 + c and ax 2 + by 2 + c have opposite signs, then they lie on the opposite sides of the line. equation of straight lines through (x 1 ) making an angle α with y = mx + c y y 1 = (x x 1 ) 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. length of perpendicular : The length P of the perpendicular from the point (x 1 ) on the line ax + by + c = 0 is given by (i) Length of perpendicular from origin on the line ax + by + c = 0 is (ii) Length of perpendicular from the point (x 1 ) on the line xcosα + ysinα = p is x 1 cosα + y 1 sinα = p Distance between Two Parallel Lines : The distance between two parallel lines ax + by + c 1 = 0 and ax + by + c 2 = 0 is Distance between two parallel lines ax + by + c 1 = 0 and kax + kby + c 2 = 0 is Distance between two non parallel lines is always zero. 6
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