STRAIGHT LINES

1 Slope of a lineIf is the angle made by a line with positive direction of x-axis in anticlockwise direction,then the value of tan is called the slope of the line and is denoted by slope of a line passing through points P(x1, y1) and Q (x2, y2) is given by2121tanyymxx = = Angle between two LINES The angle between the two LINES having slopes m1 andm2 is given by1212()tan1mmm m = +If we take the acute angle between two LINES , then tan = 12121mmm m +If the LINES are parallel, then m1 = the LINES are perpendicular, then m1m2 = Collinearity of three points If three points P (h, k), Q (x1, y1) and R (x2,y2)are such that slope of PQ = slope of QR, , 121121ykyyxhxx = or(h x1) (y2 y1) = (k y1) (x2 x1) then they are said to be Various forms of the equation of a line (i)If a line is at a distance a and parallel to x-axis, then the equation of the line isy = a.

2 (ii)If a line is parallel to y-axis at a distance b from y-axis then its equation isx = bChapter10 STRAIGHT LINES18/04/18166 EXEMPLAR PROBLEMS MATHEMATICS(iii) Point-slope form : The equation of a line having slope m and passing through thepoint (x0, y0) is given by y y0 = m (x x0)(iv)Two-point-form : The equation of a line passing through two points (x1, y1) and(x2, y2) is given byy y1 = 2121yyxx (x x1)(v)Slope intercept form : The equation of the line making an intercept c on y-axis andhaving slope m is given byy = mx + cNote that the value of c will be positive or negative as the intercept is made onthe positive or negative side of the y-axis, respectively.(vi)Intercept form : The equation of the line making intercepts a and b on x- and y-axis respectively is given by 1xyab+ =.(vii)Normal form : Suppose a non-vertical line is known to us with following data:(a)Length of the perpendicular (normal) p from origin to the line .

3 (b)Angle which normal makes with the positive direction of the equation of such a line is given by x cos + y sin = General equation of a lineAny equation of the form Ax + By + C = 0, where A and B are simultaneously not zero,is called the general equation of a forms of Ax + By + C = 0 The general form of the line can be reduced to various forms as given below:(i)Slope intercept form : If B 0, then Ax + By + C = 0 can be written asACBByx =+ or y = mx + c, where ACand =BBmc =If B = 0, then x = CA which is a vertical line whose slope is not defined and x-interceptis CA .18/04/18 STRAIGHT LINES 167(ii)Intercept form : If C 0, then Ax + By + C = 0 can be written as CCABxy+ = 1 or 1xyab+ =, where CC=and =ABab .If C = 0, then Ax + By + C = 0 can be written as Ax + By = 0 which is a linepassing through the origin and therefore has zero intercepts on the axes.

4 (iii)Normal Form : The normal form of the equation Ax + By + C = 0 isx cos + y sin = p where,2222 ABcos, sinA + BA + B = = and p = 22CA + B .Note: Proper choice of signs is to be made so that p should be always Distance of a point from a line The perpendicular distance (or simply distance)d of a point P (x1, y1) from the line Ax + By + C = 0 is given by1122 ABCA + Bxyd++=Distance between two parallel linesThe distance d between two parallel LINES y = mx + c1 and y = mx + c2 is given by1221+ccdm =. Locus and Equation of Locus The curve described by a point which movesunder certain given condition is called its locus. To find the locus of a point P whosecoordinates are (h, k), express the condition involving h and k. Eliminate variables ifany and finally replace h by x and k by y to get the locus of Intersection of two given LINES Two LINES a1x + b1y + c1 = 0 and a2x + b2y +c2 = 0 are(i)intersecting if 1122abab 18/04/18168 EXEMPLAR PROBLEMS MATHEMATICS(ii)parallel and distinct if 111222abcabc= (iii)coincident if 111222abcabc==Remarks(i)The points (x1, y1) and (x2, y2) are on the same side of the line or on the oppositeside of the line ax + by + c = 0, if ax1 + by1 + c and ax2 + by2 + c are of the samesign or of opposite signs respectively.

5 (ii)The condition that the LINES a1x + b1y + c1 = 0 and a2x + b2y + c = 0 areperpendicular is a1a2 + b1b2 = 0.(iii)The equation of any line through the point of intersection of two LINES a1x + b1y +c1 = 0 and a2x + b2y + c2 = 0 is a1x + b1y + c1 + k (ax2 + by2 + c2) = 0. The valueof k is determined from extra condition given in the Solved ExamplesShort Answer T ypeExample 1 Find the equation of a line which passes through the point (2, 3) andmakes an angle of 30 with the positive direction of Here the slope of the line is m = tan = tan 30 = 13 and the given point is(2, 3). Therefore, using point slope formula of the equation of a line , we havey 3 = 13 (x 2)orx 3y + (33 2) = 2 Find the equation of the line where length of the perpendicular segmentfrom the origin to the line is 4 and the inclination of the perpendicular segment with thepositive direction of x-axis is 30.

6 Solution The normal form of the equation of the line is x cos + y sin = p. Herep = 4 and = 30 . Therefore, the equation of the line isx cos 30 + y sin 30 = 432x + y 12= 4 or 3 x + y = 818/04/18 STRAIGHT LINES 169 Example 3 Prove that every STRAIGHT line has an equation of the form Ax + By + C = 0,where A, B and C are Given a STRAIGHT line , either it cuts the y-axis, or is parallel to or coincident withit. We know that the equation of a line which cuts the y-axis ( , it has y-intercept) canbe put in the form y = mx + b; further, if the line is parallel to or coincident with the y-axis, its equation is of the form x = x1, where x = 0 in the case of coincidence. Both ofthese equations are of the form given in the problem and hence the 4 Find the equation of the STRAIGHT line passing through (1, 2) and perpendicularto the line x + y + 7 = Let m be the slope of the line whose equation is to be found out which isperpendicular to the line x + y + 7 = 0.

7 The slope of the given line y = ( 1) x 7 is , using the condition of perpendicularity of LINES , we have m ( 1) = 1or m = 1 (Why?)Hence, the required equation of the line is y 1 = (1) (x 2) or y 1 = x 2 x y 1 = 5 Find the distance between the LINES 3x + 4y = 9 and 6x + 8y = The equations of LINES 3x + 4y = 9 and 6x + 8y = 15 may be rewritten as3x + 4y 9 = 0 and 3x + 4y 152 = 0 Since, the slope of these LINES are same and hence they are parallel to each , the distance between them is given by22159321034 =+Example 6 Show that the locus of the mid-point of the distance between the axes ofthe variable line x cos + y sin = p is 222114xyp+= where p is a Changing the given equation of the line into intercept form, we have1cossinxypp+= which gives the coordinates , 0 and 0,cossinpp , where theline intersects x-axis and y-axis, EXEMPLAR PROBLEMS MATHEMATICSLet (h, k)

8 Denote the mid-point of the line segment joining the points, 0 and0,,cossinpp Then2 cosph= and 2 sinpk= (Why?)This givescos2ph = and sin2pk =Squaring and adding we get2222144pphk+= or 222114hkp+=.Therefore, the required locus is 222114xyp+=.Example 7 If the line joining two points A(2, 0) and B(3, 1) is rotated about A inanticlock wise direction through an angle of 15 . Find the equation of the line in The slope of the line AB is 1 01 or tan 453 2 = (Why?) (see Fig.). Afterrotation of the line through 15 , the slope of the line AC in new position is tan 60 = 3 Fig. LINES 171 Therefore, the equation of the new line AC isy 0 = 3(2)x ory 32 3x+ = 0 Long Answer T ypeExample 8 If the slope of a line passing through the point A(3, 2) is 34, then findpoints on the line which are 5 units away from the point Equation of the line passing through (3, 2) having slope 34 is given byy 2 =34 (x 3)or4y 3x + 1 =0(1)Let (h, k) be the points on the line such that(h 3)2 + (k 2)2 =25(2) (Why?)

9 Also, we have4k 3h + 1 =0(3) (Why?)ork =314h (4)Putting the value of k in (2) and on simplifying, we get25h2 150h 175 =0(How?)orh2 6h 7 =0or(h + 1) (h 7) =0 h = 1, h = 7 Putting these values of k in (4), we get k = 1 and k = 5. Therefore, the coordinates ofthe required points are either ( 1, 1) or (7, 5).Example 9 Find the equation to the STRAIGHT line passing through the point of intersectionof the LINES 5x 6y 1 = 0 and 3x + 2y + 5 = 0 and perpendicular to the line 3x 5y +11 = First we find the point of intersection of LINES 5x 6y 1 = 0 and 3x + 2y +5 = 0 which is ( 1, 1). Also the slope of the line 3x 5y + 11 = 0 3is5 . Therefore,the slope of the line perpendicular to this line is 53 (Why?). Hence, the equation of therequired line is given by18/04/18172 EXEMPLAR PROBLEMS MATHEMATICSy + 1 =53 (x + 1)or5x + 3y + 8 =0 Alternatively The equation of any line through the intersection of LINES 5x 6y 1 = 0and 3x + 2y + 5 = 0 is5x 6y 1 + k(3x + 2y + 5) = 0(1)orSlope of this line is (5 3 ) 6 2kk ++Also, slope of the line 3x 5y + 11 = 0 is 35 Now, both are perpendicularso (5 3 )3 1 6 25kk + =+ork =45 Therefore, equation of required line in given by5x 6y 1 + 45 (3x + 2y + 5) =0or5x + 3y + 8 =0 Example 10 A ray of light coming from the point (1, 2) is reflected at a point A on thex-axis and then passes through the point (5, 3).

10 Find the coordinates of the point Let the incident ray strike x-axis at the point A whose coordinates be (x, 0).From the figure, the slope of the reflected ray is given bytan =35 x(1)Fig. LINES 173 Again, the slope of the incident ray is given bytan ( ) =21x (Why?)or tan =21x (2)Solving (1) and (2), we get35x =21x or x = 135 Therefore, the required coordinates of the point A are 13, 05 .Example 11 If one diagonal of a square is along the line 8x 15y = 0 and one of itsvertex is at (1, 2), then find the equation of sides of the square passing through Let ABCD be the given square and the coordinates of the vertex D be (1, 2).We are required to find the equations of its sides DC and that BD is along the line 8x 15y = 0, so its slope is 815 (Why?). The anglesmade by BD with sides AD and DC is 45 (Why?). Let the slope of DC be m. Thentan 45 =8158115mm +(Why?)