Topical Revision Worksheet 6 Coordinate Geometry Standard Graphs Student Version

Adapted from HCI Resources 1 Hwa Chong Institution Integrated Programme Secondary Three Mathematics Name: _______________________________ ( ) Class: _________ Date: _____________ Topical Revision Worksheet 6 _________________________________________________________________________________ Topic: Coordinate Geometry & Standard Graphs Question 1 [2008 EOY, Q9] marks The graph below shows part of a quadratic curve which represents the water level, y metres, of a river over a period of time in hours during the operation of a water dam nearby. At 2 pm and 8 pm, the water level was at the recommended level of 5 metres. Let x represents the time in hours since 12pm. (i) At what time did the water level reaches its minimum? Answer: at 5pm [1] (ii) Find the equation of the curve, expressing it in the form where a, h and k are real numbers. Answer: [4] (iii) Hence, or otherwise, determine the height of the water level at 12 pm. Answer: metres [2] Question 2 [2008 EOY, Q10] marks Find the range of values of k if the line intersects the curve more than once. Answer: or [5] 5 Height of water, y (in metres) 4 3 8 6 7 Time, x (hours) 2 8

Adapted from HCI Resources 2 Question 3 [2008 EOY, Q12] marks The diagram below shows a circle with radius r units and its centre is . (i) Given that the circle passes through the origin, find the exact value of r. Answer: [2] (ii) The circle is reflected about the y-axis and then moved 5 units downwards. Write down the coordinates of the new centre of the circle. Answer: [1] (iii) Find the equation of the new circle in the form where g, f and c are real constants. Answer: [3] Question 4 [2008 EOY, Q15] marks The figure below shows the graphs of and . The two curves meet at A and B. (i) Show that the coordinates of A are and find the coordinates of B. Answer: [3] (ii) Find the length of AB. Answer: or 6.36 units [2] (iii) Find the equation of the perpendicular bisector of AB. Answer: [3] y x B A x y 0

Adapted from HCI Resources 3 x y A (4, 6) B C (4, 2) D Question 5 [2009 EOY, Q1] marks Find the coordinates of the points at which the straight line intersects the curve . Answer: (−1, −4) and (−2, −3). [4] Question 6 [2009 EOY, Q9] marks The equation of a circle A is . Find the coordinates of the centre of circle A and state its radius. Answer: centre (− 3, 2), radius = 3 [3] Question 7 [2009 EOY, Q11] marks Solutions to this question by accurate drawing will not be accepted. In the figure, ABCD is a rhombus with vertices A (−4, 6) and C (4, −2). Given that the vertex B lies on the line , calculate the (i) equations of AC and BD, Answer: Equation of AC: Equation of BD: [3] (ii) coordinates of B and D, Answer: B = (2, 4) [3] (iii) equation of AB, Answer: [2] (iv) area of the rhombus ABCD and hence find the perpendicular distance from B to the line CD. Answer: units2; [4] Question 8 [2010 EOY, Q4] marks Find the range of values of k such that the curve meets the line at two distinct points. Answer: [5] Question 9 [2010 EOY, Q5] marks A quadratic curve is such that it is symmetrical about the line , where . Given that it cuts the x-axis when , and the maximum value of the curve is , (i) explain why the other x-intercept is , and [1] (ii) find, in terms of x, y and k, the equation of the curve Answer: or [3]

Adapted from HCI Resources 4 (cont’d) Given further that the distance between the two x-intercepts is 6 units, find the coordinates of the y-intercept. Answer: [3] Question 10 [2010 EOY, Q6] marks On separate axes, sketch the graphs of (i) for , and Answer: [G1] – Open downwards [G1] – Vertex in 2 nd quadrant [2] (ii) for , and Answer: [G1] – Open upwards [G1] – Vertex lies on y-axis [2] Question 11 [2010 EOY, Q7] marks The diagram below shows three small identical circles inscribed in a big circle centred at the origin. Each small circle has radius r units (where ) and one of the small circles has its centre at the point . The small circles are arranged with a rotational symmetry about the origin of order 3. (i) By considering the rotational symmetry of the diagram, explain fully why . [2] (ii) Write down the equation of the bigger circle, in terms of x, y and r. Answer: or [2] Given further that , (iii) find the coordinates of A, the point of contact between the smaller circle and the bigger circle as shown in the diagram, and Answer: [3] (iv) show that the area of the triangle ABC is units 2. [3] O x y A B C

Adapted from HCI Resources 5 Question 12 [2011 EOY, Q7] marks The variables x and y are related by the equation . A straight line passes through (5, 7) is obtained when is plotted against . (i) Given that the gradient of this line is 2, calculate the value of a and of b. Answer: and [3] (ii) Given that this line also passes through (k, 13), find the value of k. Answer: [2] Question 13 [2011 EOY, Q10a] marks A quadratic curve has a minimum point at and passes through . If the curve does not meet the line , find the range of values of m. Answer: [5] Question 14 [2011 EOY, Q14] marks Solutions to this question by accurate drawing will not be accepted. The diagram shows the quadrilateral ABCD. The coordinates of A and B are and respectively. Points C and D lie on the x-axis and (i) Find the equation of the perpendicular bisector of line AB. Answer: [3] (ii) Find the equation of AD and hence find the coordinates of D. Answer: [2] (iii) If the area of is 14 square units, find the coordinates of C Answer: [1] (iv) Find all the possible coordinates of E for which A, B, C and E are vertices of a parallelogram. Answer: or or [3]

Adapted from HCI Resources 6 Question 15 [2012 EOY, Q5] marks A set of points with coordinates lies on the graph whose equation is . The points are d units from a fixed point P. Find the coordinates of P and the value of d. Answer: [4] Question 16 [2012 EOY, Q10 c] marks Find the coordinates of the points of intersection of the line and the curve . Answer: [5] Question 17 [2012 EOY Q11] marks Solutions to this question by accurate drawing will not be accepted. The diagram shows a quadrilateral ABCD with , , , and . AB and CD intersect the x-axis at points E and F respectively. (i) Show that angle BAD = . [2] (ii) Find the equation of the perpendicular bisector of AD. Answer: [3] (iii) Given that ABCD is a cyclic quadrilateral, find the exact length of the diameter of the circle passing through A, B, C and D. Answer: [2] (iv) A point P is such that BCDP is a rectangle, find the coordinates of P. Answer: [2] (v) Find the area of the quadrilateral AEFD. Answer: 39 [2]

Adapted from HCI Resources 7 Question 18 [2013 EOY, Q13] marks (i) The equation of a circle is given by . Find the centre and the radius of the circle. Answer:Centre is (5, -1) and the radius is 5 units [2] (ii) The tangent to the circle at a point A is parallel to the straight line . Find the equation of the diameter of the circle through A. Answer: [2] (iii) The circle is then reflected on the line . State the equation of the new circle formed. Answer: