Straight Line Paper 1 Section Each correct answer in this section is worth two marks. 1. The line with equation = a + 4 is perpendicular to the line with equation 3 + + 1 = 0. What is the value of a?. 3 B. 1 3. 1 3 D. 3 4. The line through the points ( 2, 5) and (7, a) has gradient 3. What is the value of a?. 8 B. 22. 28 D. 32 2. What is the distance, in units, between the points ( 1, 2) and (4, 5)?. 8 B. 16. 34 D. 58 5. The equation of a line is 3 = a + 1 where a = 0 is a constant. Given that the line has a gradient of 7 5, what is the value of a?. 21 5 B. 7 5. 7 5 D. 21 5 3. What is the distance, in units, between the points (a, b) and ( b, a)?. 2 a 2 + b 2 B. 2(a + b) ( a ). 2 + b D. 2 a 2 + b 2 hsn.uk.net Page 1 6. The line with equation = 3 a + 4, where a = 0 is a constant, is perpendicular to the line with equation = 1 2 + 1. What is the value of a?. 6 B. 3 2. 3 2 D. 6 Questions marked c SQ
7. The line l passes through (3, 2) 10. Triangle B is shown below. and is parallel to the line with equation = 1 2 + 5. What is the equation of l? B. 2 + 1 = 0 B. 2 7 = 0. 2 + 7 = 0 D. 2 5 = 0 D Here are two statements about the line BD: I. BD is an altitude of triangle B II. BD is the perpendicular bisector of 8. Find the equation of the line passing through (6, 4) and parallel to the line with equation 2 3 1 = 0.. 2 3 24 = 0 B. 3 + 2 10 = 0. 2 16 = 0 Which of the following is true?. neither statement is correct B. onl statement I is correct. onl statement II is correct D. both statements are correct D. 2 3 18 = 0 9. Given that (1, 0) is the midpoint of ( 3, a) and B(b, 2), what are the values of a and b? a b. 2 4 B. 2 5. 2 5 D. 4 2 hsn.uk.net Page 2 Questions marked c SQ
11. Triangle B with vertices ( 4, 1), B(4, 3) and (1, 5) is shown below. B M Point M(0, 2) is the midpoint of B. What is the equation of the median through?. 3 + 2 = 0 B. 4 + 8 = 0. 4 + 2 = 0 D. 3 1 = 0 12. Triangle B with vertices (6, 7), B(7, 0) and ( 1, 2) is shown below. B The line through and B has gradient 1 4. Find the equation of the altitude through.. 4 + 11 = 0 B. 4 + 22 = 0. 4 + 31 = 0 D. 8 3 27 = 0 hsn.uk.net Page 3 Questions marked c SQ
[END F PPER 1 SETIN ] Paper 1 Section B 13. Three lines have equations 2 + 3 4 = 0, 3 17 = 0 and 3 10 = 0. Determine whether or not these lines are concurrent. 4 14. The points and B have coordinates (a, a 2 ) and (2b, 4b 2 ) respectivel. Determine the gradient of B in its simplest form. 2 15. Find the equation of the straight line which is parallel to the line with equation 2 + 3 = 5 and which passes through the point (2, 1). 3 16. rag 17. The kite BD has( vertices) (1, 8), B(0, 2), ( 3, 4) and D 21 5, k as shown in the (1, 8) diagram. (a) Determine the value of k. 5 (b) Find the area of triangle BD. 4 ( ) D 21 5, k ( 3, 4) B(0, 2) hsn.uk.net Page 4 Questions marked c SQ
18. (a) The line l 1 passes through the point (1, 10) and is perpendicular to the line with equation + 2 = 1. Find the equation of l 1. 3 (b) The line l 2 passes through the point (6, 5) and makes an angle a with the positive direction of the -ais, where tan a = 1 3. Find the equation of l 2. 2 (c) Determine the coordinates of the point of intersection of l 1 and l 2. 3 19. Triangle B has vertices (1, 2), B(8, 2) and (4, 6). D (a) Find the equation of the line through the collinear points, and D. (4, 6) 2 (b) Triangle BD is right-angled at B. Find the equation of BD. 1 (c) Find the coordinates of D. (1, 2) B(8, 2) 2 20. The equation of a straight line is 3 + a + 1 = 0, where a = 0 is a constant. Given that this line has a gradient of 1 3, find the value of a, and hence state the coordinates of the point where the line cuts the -ais. 4 21. The line is the diameter of a circle with B ling on the circumference as shown below. ( 1, 5) B(3, 7) is the point ( 1, 5) and B(3, 7). Find the equation of the chord B. 3 hsn.uk.net Page 5 Questions marked c SQ
22. The diagram below shows the right-angled triangle PQ and a circle with centre (0, 5) and diameter S. S R Q 3 6 P Find the equation of the chord RS. 5 23. Triangle B has vertices ( 3, 5), B(9, 9) and (9, 3). (a) Write down the equation of B. 1 (b) Find the equation of the altitude from. 2 (c) Find the equation of the perpendicular bisector of B. 4 (d) Find where the perpendicular bisector of B and the altitude from intersect. 2 B 24. Triangle B is shown in the diagram below. ( B 2 ) 3, 7 and lie on the -ais, and B is the point ( 2 ) 3, 7. (a) The line B has a gradient of 1 2. Find the equation of B. 2 (b) B is part of the line with equation 3 + 1 = 0. Find the length of. 2 Questions marked c SQ hsn.uk.net Page 6
25. The line with equation 3 + a + 1 = 0, where a is a constant, is perpendicular to the line with equation 2 = 2. Find the value of a. 3 26. Triangle B has vertices (4, 7), B( 2, 1) and (6, 3). (a) Find the equation of line p, the PSfrag median from. 3 (b) Find the equation of line q, the altitude B from. 3 (c) Find the point of intersection of the lines p and q. 3 27. The line with equation 4 + 3 24 = 0 intersects the -ais at P and the -ais at R. (a) Write down the coordinates of P and R. 1 (b) The perpendicular bisector of PR meets the line = 1 at Q. Find the coordinates of Q. 5 (c) Show that P, Q and R could be three vertices of a square. 3 28. Triangle PQR is shown in the diagram. P(1, 3) R Q(7, 2) The line PR has equation 6 + 9 = 0, and QR has equation 5 17 = 0. (a) Find the coordinates of point R. 2 (b) Hence find the equation of the median through R. 3 hsn.uk.net Page 7 Questions marked c SQ
29. The points (3, 2), B(2a, 12) and (a, 1) are collinear. Find the value of the constant a. 4 30. Triangle B is shown in the diagram. ( 4, 6) B(4, 3) The line has equation 8 + 2 = 20, and B has equation + 6 = 14. Find the equation of the altitude through. 5 [END F PPER 1 SETIN B] hsn.uk.net Page 8 Questions marked c SQ
Paper 2 1. Find the equation of the perpendicular bisector of the line joining (2, 1) and B(8, 3). 4 2. rag 3. rag 4. rag hsn.uk.net Page 9 Questions marked c SQ
5. rag 6. rag 7. rag 8. Find the equation of the line through the point (3, 5) which is parallel to the line with equation 3 + 2 5 = 0. 2 9. rag hsn.uk.net Page 10 Questions marked c SQ
10. The vertices of a triangle are P( 1, 1), Q(2, 1) and R( 6, 2). Find the equation of the altitude of triangle PQR, drawn from P. 3 11. Find the equation of the median D of triangle B where the coordinates of, B and are ( 2, 3), ( 3, 4) and (5, 2) respectivel. 3 12. rag 13. rag 14. Triangle B has vertices ( 1, 6), B( 3, 2) and (5, 2). ( 1, 6) Find (a) the equation of the line PSfrag p, the median from of triangle B. (5, 2) 3 (b) the equation of the line q, the perpendicular bisector of B. 4 (c) the coordinates of the point of B( 3, 2) intersection of the lines p and q. 1 Questions marked c SQ hsn.uk.net Page 11
15. rag 16. rag 17. rag hsn.uk.net Page 12 Questions marked c SQ
18. P, Q and R have coordinates (1, 2), (6, 3) and (9, 14) respectivel and are three vertices of a kite PQRS. (a) Find the equations of the diagonals of this kite and the coordinates of the point where the intersect. 7 (b) Find the coordinates of the fourth verte S. 2 19. Points (2, 3) and (8, 11) are the end points of a diagonal of rectangle BD. The diagonal BD is parallel to the -ais. Find the area of the rectangle. 5 20. The diagram shows a rhombus BD. The points B and D have coordinates (3, 5) and (9, 3) respectivel. The equation of D is = 6. B Find the size of angle D. 6 D 21. The diagram shows triangle B. circle passes through all the vertices of the triangle. is a diameter of the circle. The equation of B is 3 = 2. (a) Find the gradient of B. 3 (b) B makes an angle of a radians with the positive direction of the -ais. Find the value of a. 2 B a hsn.uk.net Page 13 Questions marked c SQ
22. The line l passes through points ( 2, 6) and B(5, 1), as shown below. ( 2, 6) P B(5, 1) The line through and P is perpendicular to l, and lies on the -ais. (a) Find the equation of the line through and P. 3 (b) Hence find the coordinates of P. 3 (c) alculate the area of the triangle P. 3 23. Find the equation of the straight line passing through the point ( 2 ) 3, 7, with a gradient of 1 2. 2 24. The line with equation + 2 6 = 0 makes an angle of φ with the -ais as shown below. + 2 6 = 0 φ alculate the value of φ correct to two decimal places. 4 hsn.uk.net Page 14 Questions marked c SQ
25. The points R(1, 2), S(5, 8) and T(11, 4) lie on the circumference of a circle. S 3 2 12 = 0 R T The line with equation 3 2 12 = 0 is the perpendicular bisector of ST. (a) Find the equation of the perpendicular bisector of RS. 4 (b) The centre of the circle is the point where the perpendicular bisectors of RS and ST intersect. alculate the coordinates of the centre of the circle. 3 26. Triangle PQR has vertices P(1, 4), Q(6, 14) and Q(6, 14) R(7, 6). (a) Find the equation of the median QS. 3 (b) Find the equation of the altitude RT. 3 (c) The median QS and the altitude RT intersect at T. R(7, 6) Find the coordinates of. S 3 P(1, 4) 27. The vertices of the triangle PQR are P(2, 6), Q( 4, 4) and R( 3, 7). (a) Find the equation of the median through R. 3 (b) Find the equation of the altitude through Q. 3 (c) The median through R and the altitude through Q intersect at point T. alculate the coordinates of T. 3 [END F PPER 2] hsn.uk.net Page 15 Questions marked c SQ