Straight Line Exercise

Page # 5 STRAIGHT LINE EXERCISE Q.1 Find the distance between the points: Q.18 A(2, 3) and B(6, 3) Q.2 Q.3 Find the distance of the point P(6, 6) from ‘ the origin. Find a point on the x-axis which is equidistant from the points A(7, 6) and B(3, 4). Q.4 Find a point on the y-axis which is equidistance from A(4, 3) and B(5, 2). Q.5 Using the distance formula, show that the points A(3, 2) B(5, 2) and C(8, 8) are collinear. Q.6 Show that the points A(7, 10), B(2, 5) and C(3, 4) are the vertices of an isosceles rightangled triangle. Q.7 Q.19 are Q.20 Q.21 Q.22 Show that the points A(5, 1), B(5, 5) and C(10, 7) are collinear. Q.10 Q.11 Q.23 Find the coordinates of the point which divides the join of A(5, 11) and B(4, 7) in the ratio 2 : 7. Q.13 Find the ratio in which the x-axis cuts the join Q.24 Q.15 Find the inclination of a line whose slope is Q.16 Find the slope of a line which passes through the points (0, 0) and (4, 2) Q.17 3 5 , find the value of x. 4 11 23 If  is the angle between the diagonals of a Show that the points A(0,6),B(2,1)and C(7,3) (ii) the coordinates of the fourth vertex. Q.25 A(1, 1), B(7, 3) and C(3, 6) are the vertices of a ABC. If D is the midpoint of BC and AL  BC, find the slope of (i) AD and (ii) AL. Q.26 Find the angle which the line joining the points (1, 3 ) and ( 2 , 6 ) makes with the x-axis. Q.27 Prove that the points A(1, 4), B(3, 2) and C(4, 5) are collinear. Also find the equations of line on which these points lie. Q.28 If A(0, 0), B(2, 4) and C(6, 4) are the vertices of a ABC, find the equations of its side. Q.29 If A(-1, 6), B(-3, -9) and C(5, -8) are the vertices of a ABC, find the equation of its medians. Q.30 Find the equation of the perpendicular bisector of the line segment whose end points are A(10, 4) and B(4, 9). If the slope of the line joining the points A(x, 2) and B(6, -8) is If  is the angle between the lines joining the points A(0, 0) and B(2, 3), and the points (i) the slope of the diagonals BD and the point of intersection. Find the slope of a line whose inclination is 30o 2 4 (iii) tan C = 3 7 are three corners of a square ABCD. Find of the points A(4, 5) and B(10, 2). Also, find Q.14 If A(1, 2), B(3, 2) and C(3, 2) be the vertices of a ABC, show that A(0, 2), B(2, 1), C(4, 0) and D(2, 3). Show that tan  = 2. Find the area of the quadrilateral whose Q.12 3 ) and (2 + 3 ). parallelogram ABCD whose vertices are Find the value of k for which the points A(-2, 3), B(1, 2), C(k, 0) are collinear. vertices are A(4,5),B(0,7),C(5,5) and D(4,2). Find the angle between the lines whose slopes C(2, -2), D(3, 5), show that tan = find the values of x and y. Q.9 1 . 3 (i) tan A = 2 (ii) tan B = D(1, y) are the vertices of a parallelogram, Find the area of ABC whose vertices are A(3, 5), B(5, 2), and C(9, 3). 3 and are (2  If the points A(-2, -1), B(1, 0), C(x, 3) and Q.8 Show that the line through the points (5, 6) and (2, 3) is parallel to the line through the points (9, 2) and (6, 5). Find the angle between the line whose slope Page # 6 STRAIGHT LINE Q.31 Find the equations of the altitudes of a ABC, whose vertices are A(2, 2), B(1, 1) and C(1, 0). Q.46 Find the equation of the line whose portion intercepted between the coordinate axes is divided at the point (, 6) in the ratio 3 : 1. Q.32 If A(4, 3), B(0, 0) and C(2, 3) are the vertices of a ABC, find the equation of the bisector of A. Q.47 Q.33 The midpoints of the sides BC, CA and AB of a ABC are D(2, 1), E(5, 7) and F(5, 5) respectively. Find the equations of the sides of ABC. A straight line passes through the point (-5,2) and portion of the line intercepted between the axes is divided at this point the ratio 2 : 3. Find the equation of the line. Q.48 If the straight line Q.34 Q.35 If A(1, 4), B(2, 3) and C(1, 2) are the vertices of a ABC, find the equation of (i) the median through A (ii) the altitude through A (iii) the perpendicular bisector of BC. Find the equation of the line through the point (1, 5) and making an intercept of 2 on the y-axis. Q.37 Find the equation of the line which is parallel to the line 2x – 3y = 8 and whose y-intercept is 5 units. Q.38 Find the equation of the line passing through the point (0, 3) and perpendicular to the line x  2y + 5 = 0. Q.39 Fi nd the equation of the line which is perpendicular to the line 3x + 2y = 8 and passes through the midpoint of the line joining the points (6, 4) and (4, 2). Q.40 Find the equation of the line whose y-intercept is -3 and which is perpendicular to the line joining the points (–2,3) and (4,–5). Q.41 Find the equation of the line passing through (3, 5) and perpendicular to the line through the points (2, 5), and (3, 6). Q.42 A line perpendicular to the line segment joining the points (1, 0) and (2, 3) divides it in the ratio 1 : 2. Find the equation of the line. Q.43 Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes, whose sum is 9. Q.44 Find the equation of the line which passes through the point (22, -6) and whose intercept on the x-axis exceeds the intercept on the yaxis by 5. Q.45 points (8, -9) and (12, -15), find the value of a and b. Q.49 Find the equation of the line for which p = 3 and  = 45o Q.50 The length of the perpendicular segment from the origin to a line is 2 units and the inclination Find the equation of the bisectors of the angles between the coordinate axes. Q.36 Find the equation of the line whose portion between the axes is bisected at the point (3, 2). 1 x y   1 passes through the a b of this perpendicular is  such that sin  =  is acute. Find the equation of the line. Q.51 1 and 3 Find the equation of the line which is at a distance of 3 units from the origin such that tan  = 5 , where  is the acute angle which 12 this perpendicular makes with the positive direction of the x-axis. Q.52 Reduce the equation y + 5 = 0 to slopeintercept form, and hence find the slope and y-intercept of the line. Q.53 Reduce the equation 3x  4y + 12 = 0 to intercepts form. Hence, find length of the portion of the line intercepts between the axes. Q.54 Reduce the equation x + y  2 = 0 to the normal form x cos  + y sin  = p, and hence find the values of  and p. Q.55 Find the values of k for which the length of perpendicular from the point (4, 1) on the line 3x  4y + k = 0 is 2 units. Q.56 Show that the length of perpendicular from the point (7, 0) to the line 5x + 12y9 = 0 is double the length of perpendicular to it from the point (2, 1). Q.57 The points A(2, 3), B(4, -1) and C(-1, 2) are the vertices of ABC. Find the length of perpendicular form C on AB and hence find the area of ABC. Page # 7 STRAIGHT LINE Q.58 What are the points on the x-axis whose x y perpendicular distance from the line   1 3 4 is 4 units? Q.59 Q.60 Q.61 Q.62 Q.63 Q.64 Q.65 Q.66 Q.67 Q.68 Q.69 Find all the points on the line x + y = 4 that lie at a uni t dist ance from th e li ne 4x + 3y = 10. A vertex of a square is at the origin and its one side lies along the line 3x  4y – 10 = 0. Find the area of the square. Find the distance between the parallel lines 4x  3y + 5 = 0 and 4x  3y + 7 = 0. Prove that the line 12x  5y  3 = 0 is midparallel to the line 12x  5y + 7 = 0 and 12x  5y  13 = 0. ANSWER KEY 10 units P(0, 2) k=7 13. (5 : 2), P(6, 0) 14. 15. 60o 1 2 1 3 19. 30° 20. 60o 16.  17. x=–2 7 4 7 (ii) (5,8) 25. (i) (ii) 3 8 3 26. 60° 27. 3x+y7=0 28. y=4, 2x3y= 0, 2x–y=0 24. (i) 29. 29x+4y+5=0, 8x–5y–21=0, 13x+14y+47=0 30. 28x 10y–19= 0 31. 2x + y  2 = 0, 3x  2y  1 = 0, x  3y + 1 = 0 32. x–3y+5=0 The perpendicular distance of a line from the origin is 5 units and its slope is -1. Find the equation of the line. 33. x  2 = 0, 6x  7y + 79 = 0, 6x + 7y + 65 = 0 Fi nd the val ue of k so that the lin es 3x  y  2 = 0, 5x + ky  3 = 0 and 2x + y  3 = 0 are concurrent. 35. x–y=0 or x + y =0 36. 7x + y + 2 = 0 37. 2x – 2y + 15 = 0 38. 2x + y  3 = 0 39. 2x 3y  7 = 0 40. 3x – 4y –12 = 0 41. 5x  y + 20 = 0 42. 3x + 9y –13 =0 43. x + 2y  6 = 0 or 2x + y  6 = 0 44. 6x + 11y – 66 = 0 or x + 2y – 10 = 0 45. 2x  3y  12 = 0 46. 2x + 5y  40 = 0 47. 3x – 5y + 25 = 0 48. a = 2, b = 3 49. 51. 52. x +y  3 2=0 50. 2 2x  y  6  0 12x + 5y –39 = 0 y = 0  x  5, m = 0 and c = 5 Find the image of the point P(1, 2) in the line x  3y + 4 = 0. Find the area of the triangle formed by the li nes x + y = 6, x  3y = 2 and 5x  3y + 2 = 0. Find the area of the triangle formed by the lines x = 0, y = 1 and 2x + y = 2. Find the area of the triangle, the equations of whose si des are y = x, y = 2x and y  3x = 4. Find the equation of the perpendicular drawn from t he ori gi n to the li ne 4x  3y + 5 = 0. Also, find the coordinates of the foot of the perpendicular. 34. Find the equation of the perpendicular drawn from the point P(2, 3) to the line x  4y + 7 = 0. Also, find the coordinates of the foot of the perpendicular. (i) 13x  y  9 = 0 (ii) 3x  y + 1 = 0 (iii) 3x  y  4 = 0 53. x y   1 , 5 units 4 3 54. xcos45° + ysin 45° = 1;  = 45°, p = 1 55. k = 2 or k = –18 58. (8, 0) and (2, 0) units, 7 sq units 5 59. (3, 1) and (–7, 11) 60. 4 sq units 61. 63. x + y + 5 2 = 0 or x +y– 5 2 = 0 6 7 k = 2 65.  ,  66. 12 sq units 5 5 64. Q.70 2. 6 2 units 3. P(3, 0) 7. x=4, y=2 8. 29 sq units 11. 60.5 sq units 12. P(3, 7) 1. 4. 10. 57. 7 2 units 5  4 3 67. 1 sq units 68. 4 sq units 69. 3x + 4y = 0,   ,   5 5  27 23  ,  70. 4x + y + 5 = 0,   17 17 