Raffles Institution 2024 Y3 SUPP WS Coordinate Geometry

Page 1 of 8 RAFFLES INSTITUTION MATHEMATICS DEPARTMENT 2024 YEAR 3 RP MATHEMATICS TOPIC 5: COORDINATE GEOMETRY (MATHS 1 & MATHS 2) SUPPLEMENTARY WORKSHEET Name: Class: 3 ( ) Date: 1 2022/Y3RP/M2/T1/Q3 The equation of a circle, 1C , with centre A , is 22 6 4 87xy xy+−−= . (i) Find the coordinates of A and the radius of 1C . [3] (ii) The point ( )11, 8P lies on 1C . Find the equation of the tangent to 1C at P . [2] (iii) Determine if the point ( )9, 7Q − lies inside 1C . Justify your answer. [2] [Ans: (i) A(3, 2), 10 units (ii) 4 68 33yx= −+ (iii) No] 2 2022/Y3RP/M2/T1/Q4 The diagram shows a right-angled triangle OBC , where O is the origin and the point C is at ( )10, 0 . OB is perpendicular to BC and the equation of BC is 2 20yx= −+ . Point D lies on BC produced such that : 7:2BD CD = . (i) Show that the coordinates of point B are (8, 4). [2] (ii) Find the coordinates of D .[2] (iii) Explain why it is possible to draw a circle passing through O , B and D . [1] (iv) Given that point A is a reflection of point B in the x- axis, find the area of triangle ACD . [3] [Ans: (ii) 4310 , 155D − (iv) 13 5 units2] 3 2021/Y3RP/M2/T1/Q2 Solution to this question by accurate drawing will not be accepted. The diagram shows an isosceles triangle ABC with vertices A( )3,1 and B( )2,3− and AB AC= . The equation of the line BC is 3 27 7 0xy+−= . Find (i) the equation of the perpendicular bisector of BC, [2] (ii) the coordinates of the mid-point, M, of BC, [3] (iii) the area of triangle ABC. [2] (iv) the coordinates of the point P which lies on AB produced such that : 2:3AB AP = . [2] [Ans:(i) 7 83yx= −+ (ii) 111 ,422   (iii) 114 2 units2(iv) 14 , 42 − ]

Page 2 of 8 4 2021/Y3RP/M2/T1/Q5 A circle passes through the points ( )3, 0A and ( )1, 8B − . The x -axis is a tangent to the circle at A. (i) Explain briefly why the x-coordinate of the centre of the circle is 3. [1] (ii) Find the equation of the circle. [3] [Ans: (ii) ( ) ( ) 22 3 52xy− +− = ] 5 2020/Y3RP/M2/T1/Q4 Solutions to this question by accurate drawing will not be accepted. The diagram shows a circle 22 2 4 20 0xy xy+++−= with centre G . TA and TB are tangents to the circle at A and B respectively from point ( )14, 8T − . (i) State the centre G of the circle. [1] (ii) Find the length of GT , leaving your answer in surd form. [2] (iii) Explain why GT is the diameter of the circle that passes through the points ,,AG B and T . Hence, find the equation of this circle. [3] [Ans: (i) ( )1, 2−− (ii) 3 29 units (iii) ( ) 2 213 261524xy − ++ = ] 6 2020/Y3RP/M2/T1/Q5 Solutions to this question by accurate drawing will not be accepted. In the diagram, ABCD is a parallelogram in which C is ( )1, 5− , D is ( )8, 4− , N is the foot of the perpendicular from C to AD and 1 5DN DA= . The equation of the line AD is 2 12 0xy++ = and it cuts the x-axis at E . Find (i) the equation of CN . [2] (ii) the coordinates of N . [2] (iii) the coordinates of A , [2] (iv) the area of quadrilateral OCDE . [3] [Ans:(i) 1 11 22yx= + (ii)( )7,2− (iii)( )3, 6−− (iv)30 units2] 7 2019/Y3RP/M1/T1/Q3 The diagram shows a line segment AB where A is the point ( )4, 2−− and B is the point ( )2,10 . (i) Find the equation of the perpendicular bisector of AB. [3] (ii) ( ),2Ck − is a point on the perpendicular bisector of AB , find the value of k. [2] (iii) Find the area of triangle ABC. [2] (iv) D is a point on the y -axis such that the area of triangle ACD is equal to the area of triangle ABC. Write down all the possible coordinates of D. [2] [Ans: (i) 17 22yx= −+ (ii) k = 11 (iii) 90 units2 (iv) ( ) ( )0,10 or 0, 14− ]

Page 3 of 8 8 2019/Y3RP/M 2/T1/Q4 A circle C1 touches the lines x = 4 and y = –2 at points A and B respectively. The centre of the circle lies on the line 3x + 2y = 0. The value of the x-coordinate of the centre is negative. (i) Show that the centre of C1 is (−4, 6). [2] (ii) Find the radius and hence write down the equation of the circle. [2] A point P lies on BA produced such that BA : BP = 2 : 3. A point H is the highest point on C1. (iii) Find the coordinates of P. [2] (iv) Find the area of triangle APH. [3] [Ans: (ii) 8 units, 22( 4) ( 6) 64xy+ +− = (iii) P(8, 10) (iv) 32 units2] 9 2018/Y3RP/M1/T1/Q5 Solutions to this question by accurate drawing will not be accepted. Two points have coordinates A (–1, –3) and C (–7, –7). B is the point on the y -axis and ABCD is a rhombus. (a) Find the equation of the perpendicular bisector of AC. [3] (b) Calculate the coordinates of B and D. [3] [Ans: (a) 3 112yx= −− (b) (0, 11)B − , ( 8,1)D − ] 10 2018/Y3RP/M2/T1/Q7 A circle C1 is given by the equation 22 8 2 3 0.xy xy+ + − −= (i) Find the coordinates of the centre and the radius of circle C1. [2] Another circle C2 of centre ( )2, 4 and radius 5 unit touches the circle C1 at point P. Find (ii) the coordinates of point P, [2] (iii) the equation of the common tangent to circles C1 and C2 at point P. [2] [Ans: (i) ( )4, 1 , 2 5 Centre Radius units= −= (ii) (0, 3) (iii) 23yx= −+ ] 11 2017/Y3RP/T1/Q3 Solutions to this question by accurate drawing will not be accepted. In the diagram, EF is parallel to HG and the coordinates of E , F and G are ( ) ( ) ( )2,10 , 8,12 and 13,7 . It is also given that H lies on the line yx= . Find (i) the equation of EF, [2] (ii) the coordinates of the point H, [2] (iii) the area of the quadrilateral EFGH. [2] [Ans: (i) 1 28 33yx= + (ii) (4, 4) (iii) 50 units2] x C1 A y= –2 y x = 4 B

Page 4 of 8 12 2017/Y3RP/T1/Q5 A circle C passes through the points ( )6,10A − and ( )3,1B − . (i) Given that the centre of C lies on the line 3 42yx= −− , find the equation of C. [5] (ii) Write down the equations of the tangents to the circle that are parallel to the x- axis. [1] [Ans: (i) ( ) ( ) 22 6 5 25xy+ +− = (ii) y = 0 and y = 10] 13 2016/Y3RP/T1/Q5 The equation of a circle is 22 6 14 17 0xy x y++− += . (i) Find the centre and radius of the circle. [3] (ii) Show that the point ( 8, 3)− lies on the circle. [1] (iii) Find the equation of the line which is a tangent to the circle at the point (̶ 8, 3).[2] [Ans: (i) ( ̶ 3, 7), 41 unites (iii) 5 74yx= −− ] 14 2016/Y3RP/T1/Q6 Solution to this question by accurate drawing will not be accepted. The diagram, which is not drawn to scale, shows a triangle ABC in which the point A is ( )7,9 and the point B is ( )1, 3−− . The point C lies on the perpendicular bisector of AB and the equation of the line AC is 8 47yx= − . Find (i) the equation of the perpendicular bisector of AB, [3] (ii) the coordinates of C. [2] The point D is such that ACBD is a rhombus, (iii) Find the coordinates of D. [2] (iv) Show that 2AB CD= . [2] (v) E is a point on CA produced such that 23CE CA= . Find the coordinates of E.[2] [Ans: (i) 2 53yx= −+ (ii) (6, 1) (iii) (0, 5) (v) 17 ,132   ] 15 2015/Y3RP/T1/Q4 The diagram, which is not drawn to scale, shows a circle with centre A and the tangent to the circle at B (2, 3)− . The centre of the circle lies on the y- axis and the equation of the tangent is 2 5 11 0xy++= . (a) Find the coordinates of A. [2] (b) Find the equation of the circle. [2] [Ans: (a) (0, −8) (b) 22 ( 8) 29xy++ = ] 16 2015/Y3RP/T1/Q5 Solutions to this ques