HCI 2023 JC1 MYE 9649 (Solutions)

1 Let nP be the statement 22 n nnab a b  for n  When 1,n  LHS = 2 ab R H S 2 ab 1 is true.P Assume kP is true for for some k  , i.e. 22 k kkab a b  Want to prove that 1kP is true, i.e. 1 11 22 k kkab a b       11 11 11 11 11 11 11 22 --- *22 22 4 244 24 4 24 2 k kk k kk kk k k kk kk k k kk kk k k kkkk kkk ab a b ab ab a b ab aa b a b b ab ab a b a b ab ab a b a b aba bab abab                                 04 k ab  Hence kP is true 1kP is true. Since 1P is true and kP is true 1kP is true, by mathematical induction nP is true for all .n   2(a) (**)

2(b) Let the polar coordinates of N be11,r  and the polar coordinates of P be 2c o s,a   1 2 OPC OC PC PON OPC CON               2 1 cos 2 cos cos 2c o s ON OP a ra      22   1     2 11 2 cos cos 2 1 1 cosra a a      (Shown) 3(a) Characteristic equation:   2 56 0 32 0 3o r 2 mm mm mm       32 nn nuA B  1ua ,  32aA B 2ub ,  94bA B N P O C C O N P

23 3 2 B ab abB    32 32 22 33 aA B aba baA     2 3 baA , 3 2 abB       1 1 23 3232 23 3 2 n n n n n n ba a bu ub a a b        3(b)         1 1 2 2 1 23 3 2 23 3 2 n n n n n n ba a bu u ba a b               1 1 1 223 3 11 223 32 3 n n n n ba a bu u ba a b                If 2ba , as n, 1 2 03 n ,    1 lim 3 n n n u u     If 2ba , as n, 1 lim 2 n n n u u     Or   2 Let the limit be 56 0 32 0 3o r 2 L LL LL LL     

4(a) The polar equation form of a conic is 1c o s kr e   . Let F be the pole and ,AB have polar coordinates  ,AF  and  ,BF  respectively Since ,AB lies on the conic, 1c o s kAF e   and 1c o s kBF e   1c o s ke AF and 1c o s ke BF  1c o s ke BF Summing, 1 cos 1 cos 2kk eeAF BF      11 2 AF BF k 4(b) B y (i), F A B O

11 1 1 1 11 2 21 ----------------(1) PF F Q k PF PFFQ k   Similarly 22 2 2 2 11 2 21 -------------(2) PF F R k PF PFFR k   Summing (1) and (2) ,  12 12 12 12 12 12 2211 22 PF PF PF PFFQ F R k k PF PF PF PFFQ F R k              Since this is an ellipse, 12PFP F is a constant, let 12PFP F C , 12 12 2 2 constantPF PF C FQ F R k   (Shown) 5 (a)    :, 0 :, 0 :, Aa B a Px y  22 22() ()AP BP x a y x a y                 2 22 22 4 22 2 22 2