DHS Polar Coordinates (9649) Topical Revision

Polar Coordinates H2 Double Math Polar Coordinates [FM] 2.

Polar Coordinates 4 The curve C1 has polar equation cos 2ra a  for 02   , where a is a positive constant. The curve C2 is obtained by rotating C1 through an angle of 2  anticlockwise about the pole. (i) Find a polar equation for C2 in terms of cos 2 . [1] (ii) On the same diagram, sketch the curves C1 and C2, indicating the polar coordinates of the points of intersections of the curves. [2] (iii) Find the exact area of the regions that lie within both C1 and C2. [4] (iv) A design for a necklace is made by taking the combined graphs of C1 and C2, but with the perimeter of the regions described in part (iii) removed. Find the perimeter of the n e c k l a c e . [ 2 ] [CJC/FM/2017/Promo/Q4] 5 Show that the gradient of tangen t of the polar curve with equation 1 1c o sr   is cot .2  [5] [EJC/FM/2017/Promo/Q8b] 6 The curve C has equation 222 2 150xx y y  . (a) Show that a polar equation for C can be expressed in the form 2 sin 2 Pr QR   , where P, Q and R are integers to be found and   . [3] (b) Hence, find the polar coordinates of the points on C which are the furthest from the pole O. [3] [JJC/FM/2017/Promo/Q2] 3.

Polar Coordinates 7 The straight line with polar equation 1 1 sin cosr ab   , is a tangent to the circle with the polar equation, 2 2c o src  , where a, b and c are real numbers, 22 0ab and 0c . By first finding the Cartesian equations of the respective polar equations, find the possible value(s) of c in terms of a and/or b. [8] [MJC/FM/2017/Promo/Q2] 8 In the above diagram, four bugs A, B, C and D are placed at the four corners of a square with side of length a. The bugs crawl counter clockwise towards the centre of the square, O, along the spiral paths. Bug A starts from the corner 22,aa . The line joining the bug A to the bug B is tangent to the path of the bug A. (i) Taking O as the origin and the coordinates of the bug A to be (, )xy , explain why the Cartesian coordinates of the bug B are (y, x) . [1] It is given that OA = r and OA makes an angle  with the x-axis. (ii) Show that the gradient of the line AB is tan 1 tan 1     . [2] (iii) By expressing d d x  and d d y  in term