3 CHAPTER. Coordinate Geometry

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1 3 CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius Cartesian Plane Ordered pair A pair of numbers a and b instead in a specific order with a at the first place and b at the second place is called an ordered pair (a, b). (a, b) (b, a) : The branch of mathematics in which geometric problems are solved through algebra by using the coordinate system, is know as coordinate geometry. Coordinate system: The position of a point in a plane is determined with reference to two fixed mutually perpendicular lines, called the coordinate axes. In the following figure: - These lines are called the coordinate axes or the axes of reference. The horizontal line X OX is called the X axis. The point O is called the origin. Convention of signs: The distances measured along OX, and OY are taken as positive and those along OX and O Y are taken as negative. Coordinates of a point in a plane. Let P be a point in a plane. Let the distance of P from the Y axis = a units and the distance of p from the x axis = b units. Then we say that the coordinates of P are (a, b) a is called the x- coordinate. The x coordinate of a point is its perpendicular distance from the Y axis measured along the x- axis and negative along the negative direction of the x axis). The x coordinate is also called the abscissa. 45

2 The Y- coordinate of a point is its perpendicular distance from the x axis measured along the y- axis (Positive along eh positive direction of the y axis and negative along eh negative direction of the y axis). They y- coordinate is also called the ordinate. In the figure above. b is called the y-coordinate or ordinate of P. Quadrant: - Let X OX and YOY be the coordinate axes. These axes divided by the plane of the paper into four region called Quadrants. The XOY, YOX, X OY and Y OX are respectively known as the first, second third, and fourth quadrants. Sings of the coordinates is various quadrants: - Region Quadrant Nature of x and y Sign of Coordinate XOY I X > O, Y > O ( +, +) YOX II X < O, Y > O (, + ) X OY III X < O, Y < O (, ) Y OX IV X > O, Y < O ( +, ) Coordinates of a point on the x axis: - Every point on the x-axis is at a distance of 0 unit from the x axis. So, its ordinate is Thus, the coordinates of every point on the y axis are of the form (x, 0). Coordinates of a point on the y-axis: - Every point on the y-axis is at a distance of unit from the y axis. So, pts abscissa is O. Thus, the coordinates of every point on the y axis are of the form (O, Y) The coordinates of the origin are (O, O) 46

3 IX ACADEMIC QUESTIONS Subjective Assignment Write down the coordinates of each points P,Q, R, S and T as show: - 2. Draw the lines X OX and YOY as axes on a plane paper and Plot the points given below: - (i) A (5, 3) (ii) B ( 3, 2 ) (iii) C ( 5, 4) (iv) D (2, 6 ) 3. In which quadrants do the given points lie: (i) (4, 2) (ii) ( 3, 7) (iii) ( 1, 2) ( iv) (3, 6) 4. On which axes do the given points lie? (i) (7, 0) (ii) (0, 3) (iii) (0, 6) (iv) ( 5, 0) 5. Draw the graph of the equation y = x Draw the graph of the equation y = 2x 3 7. Draw the graph of the equation y = 2x. 8. Draw the lines XOX and YOY as the coordinate axes on a paper and plot the following points on it. (i) P (7, 4) (ii) Q ( 5, 3) (iii) R ( 6, 3) (iv) S (3, 7) (v) A (6, 0) (vi) B (0, 9) (vii) O (0, 0) (viii) C ( 3, 3) 9. On which axis do the following points lie? (i) (7, 0) (ii) (0, 5,) (iii) (0, 1) (iv) ( 4, 0) 10. In which quadrant do the given points lie (i) ( 6, 5) (ii) ( 3, 2) (iii) (2, 9) (iv) (2, 3 ) 11. Draw the graph of the equation, y = x Draw the graph of the equation, y = 3x Draw the graph of the equation, y = 5x Draw the graph of the equation, y = 3x 47

4 15. Draw the graph of equation, y = x 16. In which quadrant do the following will lie? (i) (4,, 2) (ii) ( 3, 5) (iii) ( 2, 5) (iv) (4, 2 ) 17. Plot the following points on a graph paper: (i) (3, 4) (ii) ( 2, 3 ) (iii) ( 5, 2) (iv) (4, 3 ) 18. Plot the following points on the graph paper: - (i) P (3, 0) (ii) Q ( 4, 0) (iii) R (0, 5) (iv) S (0, 7) 19. Write down the coordinates of the following point A, B, C, and D marked: Plot the following points on the graph paper: - (i) (2, 5) (ii) (4, 3) (iii) ( 9, 7) (iv) (7, 4) (v) ( 3, 2) (vi) (7, 0) (viii) ( 4, 0) (ix) (0, 4) (x) (0, 0) 21. From the adjoining, find (i) Abscissa (ii) Ordinate (iii) Coordinates of Point P. x 22. Write the answer of each of the following questions: (i) What is the name of horizontal and the vertical lines drawn to determine the position and the vertical lines Cartesian plane? (ii) What is the name of each part of the plane formed by these two lines? (iii) Write the name of the point where these two lines intersect. 23. Determine (i) Abscissa 48

5 (ii) Ordinate (iii) Coordinates of point P in the fig. 24. Find the (i) Abscissa (ii) Ordinate (iii) Coordinates of Point P. 25. Write down the (i) abscissa (ii) ordinate (iii) co-ordinates of P, Q, R and S as given in the figure. 26. Draw x axis and y axis and mark the point p (4, 5) Q ( 4, 5) R ( 4, 5) and S (4, 5). 27. Draw a ABC where vertices A, B and C are (0, 2), (2, 2) and ( 2, 2) 28. Draw a rectangle PQRS in which vertices P, Q, R and S are (1, 4), ( 5, 4), ( 5, 3) and (1, 3) 29. Joint point A, B, C and D in which A, B, C and D are (4, 6), ( 2, 3), ( 2, 5) and (4, 4) Name the figure so, formed by joining these points. 30. In the figure find (i) Abcissa (ii) Ordinate (iii) Coordinates of point Q 31. Find the coordinates of the vertices of the square ABCD of side 2a, (i) Taking AB and AD as axes (ii) Taking the centre of the square as origin and axes parallel to the sides AB, AD. 32. Draw x-axis and y axis and mark the points A (3, 9), B (4, 7) C ( 8, 9) D ( 3, 5) E (4, 2) F (7, 5) 33. Find out the quadrants in which the following points will lie: - (i) T ( 7, 6) (ii) Q (7, 3) (iii) ( 3, 9) (iv) R ( 8, 7) (v) S (3, 5) (vi) Z ( 8, 13) 49

6 34. Draw a triangle PQR whose vertices are P (1, 6) Q (7, 4) R ( 4, 4) 35. What will be the coordinates of vertices of an isosceles right triangle, right-angled at the origin with hypotenuse of 32 on. 36. Draw a triangle whose vertices are A, B and C are ( 3, 0), (3, 3) and ( 3, 3) 37. Plot and join vertices A, B, C and D (4, 3), (4, 2) ( 7, 2) and ( 7, 3) respectively of a figure. Name the figures so formed. 38. Draw a rectangle KLMN such that its vertices K, L, M and N are (5, 0), (5, 3), (0, 3) and (0, 0). 39. Construct a square ABCD such that its vertices A, B, C and D are (1, 2), ( 4, 2), ( 7, 6) and (1, 6). 40. Construct a square PQRS whose vertices P, Q, R and S are (0, 0), ( 4, 0), ( 4, 4) and (0, 4) 41. Draw a parallelogram ABCD whose vertices A, B, C and D are ( 4, 8) ( 4, 2), (0, 5) and (6 1) 42. Construct a trapezium PQRS in which vertices P, Q, R and s are (3, 0) (7, 9), ( 6, 9) and ( 2, 0) 43. Draw a parallelogram ABCD in which A ( 3, 2) B ( 5, 4) C ( 2, 4) S (0, 2). 44. Construct a trapezium PQRS in which vertices P, Q, R and S are (3, 0), (7, 9) ( 6, 9) and ( 2, 0) 45. Draw a rhombus ABCD whose vertices A, B, C and D are (1, 4.5) ( 1, 0) (1, 4.5) and (3, 0) 46. Plot four point (x, y) such that y = x, and then join these points. 47. Plot four point (x, y), such that y = 2x + 1 and then join then to have a graph of linear equation, y = 2x Draw the graph of linear equation 4x + y + 1 = Draw the graph of linear equation 8x 3y + 4 = Draw graph of y = 2x 50

7 XI SCIENCE & DIP. ENTRANCE Multiple Choice Questions Assignment 2 1. A pair of numbers a and b listed in a specific place and b at the second place is called: (a) Specific Pair (b) Pair (c) Ordered pair (d) None of these 2. The branch of mathematic in which geometric problems are solved through algebra by using coordinate system, is known as (a) Coordinate system (b) Geometry (c) Coordinate geometry (d) none 3. Coordinate axes are also known as (a) Axes of geometry (b) Pair axes (c) Axes of preference (d) Axes of reference. 4. The position of a point in a plane is determined with reference to two fixed mutually perpendicular lines. Called the: - (a) Coordinate axes (b) Axes of reference (c) Both a & b (d) None of these 5. The horizontal axis on the coordinate plane is called the (a) Y axis (b) X axis (c) Origin (d) None of these 6. The vertical axis of the coordinate plane is called the (a) Y axis (b) Origin (c) Pair (d) None of these 7. The point of intersection of both the axis is called (a) Origin (b) Intersecting (c) Both a & b (d) None of these 8. X-coordinate of a point is called the (a) Ordinate (b) Point (c) Abscissa (d) All of these 9. Y coordinate of a point is called the (a) Abscissa (b) Ordinate (c) Coordinate (d) None of these 10. The axes divided the plane of the paper into four region. Called: - (a) Plane cut (b) Region (c) Coordinate plane (d) Quadrant 11. The nature of x and y in quadrant I is: - (a) x < O, y > O (b) x >O, y < O (c) x > O, y > 0 (d) x = O, y = The sign of (x, y) coordinate is 1 st quad is: (a) (+, +) (b) (, ) (c) (+, ) (d) (+, ) 13. The coordinate (2, 4) of a pair will lie in quadrant: (a) IV (b) II (c) III (d) I 14. Which of the following will like in quadrant I: - (a) (8, 9) (b) (6, 10) (c) (4, 2) (d) All of these 51

8 15. The nature of x and y is quadrant IV is: - (a) x > 0, y < 0 (b) x < 0, y < 0 (c) x = 0, y < 0 (d) x < 0, y > Which of the following will lie in IV quadrant: - (a) (21, 2) (b) ( 4, 2) (c) (4, 2) (d) ( 4, 2) 17. Which of the following will lie in Iv quadrant: - (a) (, +) (b) (, ) (c) ( +, + ) (d) ( +, ) 18. Point C (8, 1) will lie in quadrant: (a) III (b) IV (c) II (d) None of these 19. What is the nature of x and y in quadrant II: (a) x < 0, y < 0 (b) x > 0, y > 0 (c) x < 0, y > 0 (d) x = 0, y = Which of the following will lie in II: - (a) x < 0, y < 0 (b) x > 0, y > 0 (c) x < 0, y > 0 (d) x = 0, y = Sign of (x, y) coordinate of II quadrant is : (a) (+, +) (b) (+, ) (c) (, ) (d) (, +) 22. The point Q ( 1, 1) will lie in quadrant: - (a) I (b) III (c) II (d) IV 23. The nature of x and y in quadrant III is: (a) x > 0, y < 0 (b) x < 0, y < 0 (c) x > 0, y > 0 (d) x = 0, y Which of the following lie in III quadrant (a) ( 8, 4) (b) (2, 2) (c) (9, 8) (d) (0, 2) 25. Sign of (x, y) coordinates of III quadrant is: - (a) ( +, +) (b) (, ) (c) (, + ) (d) ( +, ) 26. The point A ( 2, 1) will lie in quadrant: (a) II (b) IV (c) III (d) None of these 27. The point (x, 0) will lie on (a) y axis (b) x axis (c) In 1 st quadrant (d) Anywhere 28. The point (x, y) will lie on (a) Y axis (b) x axis (c) Both x & y (d) None of these 29. Which of the following is true for origin: (a) x axis (b) x axis (c) x = 0, y = 0 (d) All of these 30. The coordinates of origin are (a) (0, 5) (b) (0, 9) (c) (5, 5) (d) None of these 52

9 ANSWER Assignment 1 1. p(2, 4), Q(4, 2), R( 2, 3), S(5, 3), T( 4, 1) 2. IV, II, III, I 3. (i) and (iv) are on x-axis, (ii) and (iii) are on y-axis 8. II, III, IV, I 14. I, II, III, IV 17. A(2, 3), B( 4, 5), C( 5, 6), D(3, 5) 19. (i) 3 (ii) 4 (iii) (3, 4) 20. (i) x-axis and y axis (ii) I Quadrant, II Quadrant, III Quadrant, IV Quadrant (iii) Origin 21. (i) a (ii) b (iii) ( a, b) 22. (i) b (ii) a (iii) (b, a) 23. For P, x = 2, y = 8 and coordinate (2, 8) For Q, x = 2 y = 9 and coordinate ( 2, 9) For R, x = 5 y = 3 and coordinate ( 5, 3) For S, x = 5, y = 1, and coordinate (5, 1) 28. (i) 2 (ii) 5 (iii) (2, 5) 29. (i) A(0, 0),B(2a, 0), C(2a, 2a) and D(0, 2a) (ii) A( a, a), B(a, a), C(a, a) D( a, a) 31. (i) II (ii) IV (iii) III (iv) III (v) I (vi) III 33. (, 0) (4, 0) and (0, 4) Assignment 2 1.c 2.c 3.d 4.c 5.b 6.a 7.a 8.c 9.b 10.d 11.c 12.a 13.d 14.d 15.a 16.a 17.d 18.b 19.c 20.c 21.d 22.c 23.b 24.a 25.b 26.c 27.b 28.d 29.c 30.d 53

Exercise (3.1) Question 1: How will you describe the position of a table lamp on your study table to another person?

Class IX - NCERT Maths Exercise (3.1) Question 1: How will you describe the position of a table lamp on your study table to another person? Solution 1: Let us consider the given below figure of a study