Coordinate Geometry. Topic 1. DISTANCE BETWEEN TWO POINTS. Point 2. The distance of the point P(.x, y)from the origin O(0,0) is given by

Size: px

Start display at page:

Download "Coordinate Geometry. Topic 1. DISTANCE BETWEEN TWO POINTS. Point 2. The distance of the point P(.x, y)from the origin O(0,0) is given by"

Agatha Jenkins
5 years ago
Views:

1 Topic 1. DISTANCE BETWEEN TWO POINTS Point 1.The distance between two points A(x,, y,) and B(x 2, y 2) is given by the formula Point 2. The distance of the point P(.x, y)from the origin O(0,0) is given by Point 3. A.When point is on x-axis then it is (x,0). Coordinate of x is abscissa B. When point is on y-axis then it is (y,0). Coordinate of y is ordinate. 1. Find the distance between the points P( 4, 7) and Q(2, - 5) Find the distance between the points: (i) A(9,3) and B(15,11) (ii) A(7, -4) and B(-5,1) (iii) A(-6, -4) and B(9, -12) (iv) A(l, -3) and B(4, -6) (v) P(a + b, a - b) and Q(a - b, a + h). ((i) 10 units (ii) 13 units (iii) 17 units (iv) 3 2 units (v) 2 2b units 3. Find the distance of the point P(6, - 6) from the origin Find the distance of each of the following points from the origin: (i) A(5,-12) (ii)b(-5,5) (iii)c (4,-6) ((i) 13 units (ii) 5 2 units (iii) 2 l3 units) Find value of coordinate when distance is given 5. Find all possible values of a for which the distance between the points A(a, -1) and B(5,3) is 5 units. a = 8 or a =2 6. Find the ordinate of a point whose abscissa is 10 and which is at a distance of 10 units from the point P(2, -3). 3 or-9 7. If A(6, -1), B(l, 3) and C(k, 8) are three points such that AB = BC, find the value of k. k=5 or k=-3 8. If the point A(a,2) is equidistant from the points B(8, -2) and C(2, -2), find the value of a. a =5 9. Find the values of k for which the distance between the points A{k,-5)and B(2, 7) is 13 units. (7,-3) Distance from axis 10. Find the point on the x-axis which is equidistant from the points A(-2, 5)and B(2,-3). (-2, 0) 11. Find the point on the y-axis which is equidistant from the points A(-4,3)and B(5,2). (0,-2) 12. If P(x, y)is a point equidistant from the points 71(6, -1) and B(2,3), find the relation between x and y. x-y = if the point P(x, y) is equidistant from the points A(5,1) and B(-l, 5), prove that 3x = 2y. 14. Find points on the x-axis, each of which is at a distance of 10 units from the point A(11, -8). 15. Find the points on the y-axis, each of which is at a distance of 13 units from the point (-5, 7).

2 When Relation is given 16. Find the coordinates of the point equidistant from three given points A(5,1), B(-3, - 7) and C(7, -1). ( Let the required point be P(x, y). Then, PA = PB = PC => PA 2 = PB 2 = PC 2 => PA 2 = PB 2 and PB 2 = PC 2 ) 17. Prove that the points A(-3,0), B(l, -3) and C(4,1) are the vertices of an isosceles right-angled triangle. Find the area of this triangle. (show that AB 2 + BC 2 = AC 2. Key point is square of lagest side = sum of sqare of other 2 side ) 18. Prove that the points A(a, a), B(-a, - a) and C(- 3a, 3a) are the vertices of an equilateral triangle. Calculate the area of this triangle. topic 2. PROPERTIES OF VARIOUS TYPES OF QUADRILATERALS A quadrilateral is a (i) rectangle if its opposite sides are equal and the diagonals are equal. (ii) square if all its skies are equal and the diagonals are equal. (iii) parallelogram if its opposite sides are equal. (iv) parallelogram but not a rectangle if its opposite sides are equal and the diagonals are not equal. (v) rhombus but not a square if all its sides are equal and the diagonals are not equal. 1. Prove that the points A(l, - 3), B(13,9), C(10,12) and D(-2,0) taken in order are the angular points of a rectangle. Find the area of this rectangle. 2. Show that the points A(l, 2), B(5,4), C(3,8) and D(-l, 6) are the vertices of a square. 3. If P(2, -1), Q(3,4), R(-2,3) and S(-3, - 2) he four points in a plane, show that PQRS is a rhombus but not a square. Find the area of the rhombus. 4. Find the coordinates of the centre of a circle passing through the points A(2,1), B(5, -8) and C(2, -9). Also, find the radius of this circle. (Let P(x, y) be the centre of the circle, PA = PB = PC => PA 2 = PB 2 = PC 2 => PA 2 = PB 2 and PB 2 = PC 2.). 5. *If two vertices of an equilateral triangle be O(0,0) and A(3, 3), find the coordinates of its third vertex. (Let B(x, y) be its third vertex. Then, OA=OB = AB => OA 2 = OB 2 = AB 2 => OA 2 = OB 2 and OB 2 = AB 2.) 6. The two opposite vertices of a square are (1, - 6) and (5,4). find the coordinates of the other two vertices. ( Let B(x, y)be its third vertex. Then,,AB = BC=> AB 2 = BC 2, From right A ABC, we have AB 2 + BC 2 = AC 2 ) 7. If the points A(6,1), B(8, 2),C(9, 4) and D(p, 3) are the vertices of a parallelogram, find the value of p. (p = 7) Topic3 COLLINEAR POINTS Three points A, B, C are said to be collinear if they lie on the same straight line. TEST FOR COLLINEARITY OF THREE POINTS In order to show that three given points A, B, C are collinear, we find distances AB, BC and AC. If the sum of any two of these distances is equal to the third distance then the given points are collinear. 1. Prove that the points A(l, 1), B(-2,7) and C(3, -3) are collinear. 2. Show that the following points are collinear A(6,9),B(0,l)andC(-6,-7). 3. Show that the following points are collinear A(-1, -1),B(2, 3) and C(8,11).

3 4. Show that the following points are collinear P(l, 1), Q(-2,7) and R(3, -3). 5. Show that the following points are collinear P(2,0), Q(11, 6) and R(-4, -4). EXERCISE 1. Find the coordinates of the point equidistant from three given points A{5, 3), B(5, -5) and C(l, -5). (P(3,-1)) 2. By distance formula, show that the points (1, -1), (5, 2) and (9, 5) are collinear. 3. Show that the points A(-3, 2), B(-5, -5), C(2, -3) and D(4, 4) are the vertices of a rhombus. Find the area of this rhombus. (45 sq units) 4. Show that the points A(3, 0), B(6, 4) and C(-l, 3) are the vertices of a right-angled triangle. Also prove that these are the vertices of an isosceles triangle. ( 24 sq units) 5. Prove that the points A(7,10), B(-2,5) and C(3, -4) are the vertices of an isosceles right triangle. 6. Show that the points A(-5, 6), B(3, 0) and C(9, 8) are the vertices of an isosceles right-angled triangle. Calculate its area. ( 50 sq units ) 7. Show that the points O(0, 0), A(3,73) and B(3, - 3) are the vertices of an equilateral triangle. Find the area of this triangle. ( 3 13 sq. unit) 8. Show that the points A(2,1), B(5, 2), C(6, 4) and D(3, 3) are the angular points of a parallelogram. Is this figure a rectangle? ( no) 9. Show that the following points are the vertices of a rectangle: (1)A(0, -4), B(6, 2), C(3,5) and D(-3, -1) (ii) A(2,-2), B(14,10), C(ll, 13) and D(-l, 1) (iii) A(-4, -1), B(-2, -4),C(4, 0) and D(2, 3) 10. Show that the following points are the vertices of a square: A(6, 2), B(2,1), C(l, 5) and D(5, 6) (ii) P(0, -2), Q(3,1), R(0, 4) and S(-3,1) (iii) A( 3, 2), B(0,5), C(-3, 2) and D(0, -1) SECTION FORMULA The coordinates of the point P(x, y) which divides the line segment joining A(x v y 0)and B(x 2, y 2) internally in the ratio m: n are given by Midpoint formula The coordinates of the midpoint M of a line segment AB with end points A(x 1,y 1) and B(x 2,y 2) are 1. Find the coordinates of the point which divides the line segment joining the points A (4, -3) and B(9,7) in the ratio 3 : 2. (7:3) 2. Find the coordinates of the midpoint of the line segment joining the points A(-5,4)and B(7,-8). (1,-2) 3. Find the coordinates of the points which divide the line segment joining the points (-2,0) and 0,8) in four equal parts. (-3/2,2),(-1,4)(-1/2,4) 4. Find the coordinates of the points which divides the join of A(-l, 7) and 8(4; - 3) in the ratio 2 : 3. (1,3) 5. Find the coordinates of the points which divides the join of P(-5,11) and Q(4, - 7) in the ratio 7 : 2. (2,-3) 6. Find the coordinates of the points of trisection of the line segment AB, whose end points are A(2,1) and B(5, - 8).(3,-2) 7. Find the coordinates of the points which divide the join of A(-4,0) and B(0,6) in three equal parts. (-8/3,20 (-4/3,4)

4 When ratio is taken as k:1 8. In what ratio does the point P(2, -5) divide the line segment joining A(-3, 5) and B(4,-9)? (5:2) 9. Find the ratio in which the point P(m, 6) divides the join of A{ 4,3) and B(2,8). Also, find the value of m. (3:2) 10. Find the coordinates of the points of trisection of the line segment joining the points (4, -1) and (-2, - 3). (2,- 5/3),,(0,7/3) 11. The line segment joining the points (3, - 4) and (1,2) is trisected at the points P(p, - 2) and (5/3, q Find the values of p and q. (7/3,0) 12. In what ratio is the line segment joining the points A(6,3) and B(-2, -5) divided by the x-axis? Also, find the coordinates of the point of intersection of AB and x-axis. {Ratio is 3:5 and point ( 3,0)} 13. In what ratio is the line segment joining the points (-2, - 3) and (3,7) divided by the y-axis? Also, find the coordinates of the point of division. (2:3) and (0,1) 14. Find the ratio in which the line 2x + y-4=0 divides the line segment joining A(2, - 2) and B(3, 7). ( 2:9). 15. The point P divides line segment joining the points (-1, 3) and (9,8)in ratio of K:1.if p lies on the x-y+2=0, find the value of k. (2/3) 16. The point P divides line segment joining the points (-1, 3) and (9,8)in ratio of 1:3.if p lies on the 2x-y+k=0, find the value of k. (-4) 17. The line joining the points 71(2,1) and B(5, - 8) is trisected at the points P and Q. It the point P lies on the line 2x-y + k = 0, find the value of k. (K=-8) 18. The line joining the points 71(4, -5) and B(4,5) is divided by the point P such that- AP/AB = 2/5 Find the coordinates of P. (2,2) 19. Find the coordinates of the midpoint of the line segment joining: (i) 71(3,0)and B(5,4), (ii) P(-ll, -8) and Q(8, -2). (4,2) 20. The coordinates of the midpoint of the line segment joining the points A(2p+1,4) and B(5, q- l)are (2p,q). Find the values of p and q. P=3,Q=3 21. The midpoint of the line segment joining 7l(2fl,4) and B(-2,3b) is M(l, 2a +1). Find the values of a and b. a=2, b=2 22. The line segment joining 7l(-2,9) and B(6,3) is a diameter of a circle with centre C. Find the coordinates of C.(2,6) 23. AB is a diameter of a circle with centre C(-l, 6). If the coordinates of A are (-7,3), find the coordinates of B.5, Find the lengths of the medians of a Δ ABC whose vertices are A(7, -3), B(5,3) and C(3, -1). 26. D(3, - 2), E(-3,1) and F(4, - 3) be the midpoints of the sides BC, CA and AB respectively of A ABC. Then, find the coordinates of A, B, C. ( A(-2,0), B(10, - 6) and C(-4,2).) ( the diagonals of a parallelogram bisect each other. Therefore, O is the midpoint of AC as well as that of BD.) 27. In what ratio does the point P(2, 5) divide the join of 71(8, 2) and B(-6,9)?3:4 28. (i) Find the ratio in which the point P(-6, a) divides the join of A(-3, -1) and B(-8,9). Also, find the value of a. 3:2,a=5 (ii) Find the ratio in which the point (-3, k) divides the line segment joining the points (-5, -4) and (-2,3). Hence, find the value of K.2:1, k=2/3 29. Find the ratio in which the point P(n, 1.) divides the join of/l(-4, 4) and B(6, -1) and hence find the value of a. 3:2, A=2 30. In what-ratio is the line segment joining A (2, -3) and 8(5, 6) divided by the x-axis? Also, find the coordinates of the point of division.(1:2), (3,0) 31. In what ratio is the line segment joining the points A(-2, - 3) and 8(3, 7) divided by the y-axis? Also, find the coordinates of the point of division. 2:3), (0,1) 32. if (-2, -1), (a, 0), (4, b) and (1,2) arc the vertices of a parallelogram, find the values of a and b. a = 1 and b = The three vertices of a parallelogram ABCD, taken in order, are A(l, -2), B(3,6) and C(5,10). Find the coordinates of the fourth vertex D. D(3,2) 34. The coordinates of one end point of a diameter AB of a circle are A(4, -1) and the coordinates of the centre of the circle are C(l, -3). Find the coordinates of B. B(-2, -5). (clearly C is the midpoint of AB.)

5 CENTROID OF A TRIANGLE The point of intersection of the medians of a triangle is called its centroid. 35. Find the centroid of Δ ABC whose vertices are A(-3,0), B(5,-2) and C(-8, 5). G(-2,1) 36. Two vertices of a Δ ABC are given by A(6,4) and B(-2,2), and its centroid is G(3,4). Find the coordinates of the third vertex C of ΔABC. 37. If A (5, -1), B(-3, -2) and C(-l, 8) are the vertices of ABC, find the length of the median through A and the coordinates of the centroid /3,5/3 38. Find the lengths of the medians of a ABC having vertices at A(0,-1), B(2,1) andc(0, 3).AD = 10,BE =2, CF = Find the coordinates of the centroid of a ABC whose vertices are A (6, -2), B(4, -3) andc(-l,-4) Find the centroid of ABC whose vertices are A(-1, 0),B(5, -2) and C(8, 2). 4,0 Find missing coordinate when centroid is given 1. A(3, 2) and B(-2,1) are two vertices of a ABC, whose centroid is G(5/3, -1/3) Find the coordinates of the third vertex C. (4-4) 2. If G(-2,1) is the centroid of a ABC and two of its vertices are A(l, -6) and B(-5, 2), find the third vertex of the triangle.- (2,7) 3. Find the third vertex of a ABC if two of its vertices are B(-3,1) and C(0, -2), and its centroid is at the origin. (3,1) DIAGNALS OF PARALLELOGRAM BISECTS EACH OTHER 4. Show that the points a.(3,1), B(0, -2), C(l, 1) and D(4, 4) are the vertices of a parallelogram ABCD. 5. If the points P(a, -11), Q(5, b), R(2,15) and S(l, 1) are the vertices of a parallelogram PQRS, find the values of a and b. (a=4,b=3) 6. If three consecutive vertices of a parallelogram ABCD are A(1, -2), B(3, 6) and C(5,10), find its fourth vertex D.3,2 7. In what ratio does the line x - y - 2 = 0 divide the line segment joining (3, -1) and (8, 9)? 2:3 8. Determine the ratio in which the line 3x + 4y - 9 = 0 divides the line segment joining the points 74.(1, 3) and B(2,7).

6 AREA OF A TRIANGLE The area of a A ABC with vertices A{x 1,y 1), B(x 2, y 2) and C{x 3, y 3) is given by CONDITION FOR COLLINEARITY OF THREE POINTS 1. Find the area of the triangle whose vertices are A(2, 7), B(3, -1) and C(-5,6) sq units. 2. Find the area of the triangle whose vertices are (i) A(1, 2), B(-2, 3) and C(-3, -4) (ii) A(2,4), B(-3, 7) and C(-4, 5) (iii) P(10, -6), Q(2,5) and R(-l, 3) (iv) P(4,4), Q(3, -16) and R(3, -2) (11, 6.5,24.5, 7 sq units) 3. Find the area of quadrilateral ABCD D(4, 5) whose vertices are A(-5,7), B(-4, - 5), C(-l, - 6) and D(4,5). (72 sq units.) (Join A and C. Then, area of quad. ABCD =(area of A ABC) + (area of A ACD).) 4. (i) Find the area of a quadrilateral ABCD whose vertices are A(-4, -2), B(-3, -5),C(3, -2) andd(2, 3). 28 sq units (ii) Find the area of the quadrilateral whose vertices are A(0, 0), B(6, 0), C(4, 3) and D(0, 3). 15 sq units 5. Find the value ofk for which the area formed by the triangle with vertices A(k, 2k), B(-2,6) and C(3,1) is 5 square units. 2,2/3 6. Show that the points A(-l, 1), 6(5, 7) and C(8,10) are collinear. 7. Show that the points A(a,b + c), B{b,c + a) and C{c,a+b) are collinear. 8. Find the value of p for which the points (-5,1), (1, p) and (4, -2) are collinear. p = Show that the following points are collinear: 10. (i) A(0,1), B(l, 2) and C(-2, -1) (ii) A(-5,1), B(5,5) and C(10, 7) (iii) P(a, b + c), Q(b, c + a) and R(c-, a + b) 11. Find the value of p for which the points A(-l, 3), B(2, p) and C(5, -1) are collinear. P=1 12. Find the value of k for which the points A(3, 2), B(4, k) and C(5, 3) are collinear. K=5/2 13. For what value of p are the points A(-3, 9), B(2, p) and C(4, -5) collinear? P= For what value of x are the points A(-3,12), B(7,6) and C(x, 9) collinear?2 15. For what value of y are the points P(l, 4), Q(3, y) and R(-3,16) collinear? If the vertices of a A ABC are 71(1, k), B(4, - 3) and C( -9, 7) and its area is 15 sq units then find the value of k.-3, 21/ Find the area of a rhombus whose vertices taken in order are the points 71(3, 0), B(4,5), C(-l, 4) and D(-2, -1). 24 sqare units ( area = ½ d1d2) 18. If the points A(x, y), 8(-5, 7) and C(-4,5) are collinear then show that 2x + y + 3 = 0.

11 2 marks

14 3 marks

Chapter 7 Coordinate Geometry

Chapter 7 Coordinate Geometry 1 Mark Questions 1. Where do these following points lie (0, 3), (0, 8), (0, 6), (0, 4) A. Given points (0, 3), (0, 8), (0, 6), (0, 4) The x coordinates of each point is zero.