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 terms of r,  and d d r  , show that d d r r  . [4] (iv) It is known that the polar equation of the path of the bug A is in the form erk  , show that 42 2 ake   , assuming the pole is at the centre of the square. [2] (v) Find the exact distance travelled by the bug A for 24    . [4] [SAJC/FM/2017/Promo/Q10] A B C D O

Polar Coordinates 9 A curve C has polar equation given by 22 2 cos sinra b   , where a and b are non-zero constants. (a) State the condition(s) on a and b so that C has tangent(s) through the pole. [1] (b) State the condition(s) on a and b so that C will never come back to the pole. [1] (c) Sketch C, where 02   , for the following cases: (i) ab , [2] (ii) a > 0 and b < 0. [2] (d) Given that a = 4 and b = 2, find the exact area of the region enclosed by C. [3] (e) Write down the polar equation of the curve if C is (i) rotated 90 anti-clockwise about the pole, [1] (ii) reflected about the line y = x. [1] [TJC/FM/2017/Promo/Q8] 10 Sketch the curve sin 2 , for 0, 0 2 .rr     [2] Describe the curves sin 2 , where 0, rn r n  are positive integers and show that the area enclosed by such a curve is independent of n. [6] [VJC/FM/2018/P1/4] 11 The curve D has polar equation 16sin 2r  , where 02 π . (i) Sketch D, indicating clearly all key features and symmetries of the curve. [2] (ii) Find the arc length of D. [2] The locus of points  ,r  satisfying 16sin 32 r  forms a region R. (iii) Find the exact area of R. [5] [NJC/FM/2018/P2/4] 12 The curve  has polar equation 1c o s, 0 2 .r     The circle with equation 22(2 ) 4xy intersects  at O, A and B, where O is the pole. Determine the perimeter of the sector OAB, where OA and OB are straight line segments and AB is an arc on  that lies within the circle. Leave your answer in an exact surd form. [8] [VJC/FM/2019/MCT/6]

Polar Coordinates 13 The curve 1T has polar equation 32 c o s 3r  , 02   . (i) Sketch 1T , indicating all the key features and equations of lines of symmetry of the curve. [3] (ii) A piece of wire of length 32 units long is used to bend into the shape of 1T . State, with a reason, whether the wire is long enough to do so. [2] Another curve 2T has polar equation 2c o s 3r  , 02   . (iii) Use calculus to evaluate the exact area enclosed in between 1T and 2T . [3] [RI/FM/2019/MCT/3] 14 [It is given that 3cos3 4cos 3cos  .] A polar curve C has equation sin 2 where 0 2r   . O is the pole and the tangent at a point (, )P r  on the curve is parallel to the line 2   . (i) Sketch C. [2] (ii) Show that 2cos 3  at P. [3] (iii) Find the area bounded by C, the line  = 0 and the tangent at P. [5] [TJC/FM/2018/P1/6] 15 (a) The Archimedean spiral, S, which was first studied by the Greek mathematician Archimedes in the 3rd century BC, has polar equation given by ra b  where a and b are non-negative real constants and 0  . The Archimedean spiral has the property that any ray from th e pole intersects successive turnings of the spiral at points with a constant separation distance d, hence also the name “arithmetic spiral”. (i) Prove the above property and state the value of d. [2] (ii) For the case when a = b, prove that the angle which the tangent to S at the point  ,r  makes with the initial line is given by   1 tan 1     and hence write down the cartesian equation of the tangent to S at the point where 0  . [5]

Polar Coordinates (b) Let A and B be two points on a polar curve corresponding to and    respectively. The area of the curved surface generated when the arc AB is rotated completely about the initial line is given by the integral 1 2 2 2 d2π sin d d rrr        . Use the above integral to derive the formula for the surface area of a sphere of radius a, explaining your working clearly. [3]