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2 physicsadmathstuto.com Jauay 009 blak 3. The ectagula hypebola, H, has paametic equatios x = 5t, y = 5 t, t 0. (a) Wite the catesia equatio of H i the fom xy = c. Poits A ad B o the hypebola have paametes t = 1 ad t = 5 espectively. (b) Fid the coodiates of the mid-poit of AB. (3) 6 *N34694A068*

3 physicsadmathstuto.com Jauay A paabola has equatio y = 4ax, a > 0. The poit Q (aq, aq) lies o the paabola. blak (a) Show that a equatio of the taget to the paabola at Q is yq = x + aq. This taget meets the y-axis at the poit R. (b) Fid a equatio of the lie l taget at Q. which passes though R ad is pepedicula to the (3) (c) Show that l passes though the focus of the paabola. (d) Fid the coodiates of the poit whee l meets the diectix of the paabola. () 16 *N34694A0168*

4 physicsadmathstuto.com Jue The paabola C has equatio y = 16x. blak (a) Veify that the poit P(4t, 8t) is a geeal poit o C. (b) Wite dow the coodiates of the focus S of C. (c) Show that the omal to C at P has equatio y + tx = 8t + 4t 3 (5) The omal to C at P meets the x-axis at the poit N. (d) Fid the aea of tiagle PSN i tems of t, givig you aswe i its simplest fom. 16 *M35146A0164*

5 physicsadmathstuto.com Jauay y y =1x blak B P A S x Figue 1 Figue 1 shows a sketch of pat of the paabola with equatio y =1x. The poit P o the paabola has x-coodiate 1 3. The poit S is the focus of the paabola. (a) Wite dow the coodiates of S. The poits A ad B lie o the diectix of the paabola. The poit A is o the x-axis ad the y-coodiate of B is positive. Give that ABPS is a tapezium, (b) calculate the peimete of ABPS. (5) 8 *N35143A084*

6 physicsadmathstuto.com Jauay 010 blak 7. The ectagula hypebola H has equatio xy= c, whee c is a costat. The poit P ct, c is a geeal poit o H. t (a) Show that the taget to H at P has equatio t y+ x=ct The tagets to H at the poits A ad B meet at the poit (15c, c). (b) Fid, i tems of c, the coodiates of A ad B. (5) 14 *N35143A0144*

7 physicsadmathstuto.com Jue The paabola C has equatio y = 0 x. blak (a) Veify that the poit P(5 t,10 t ) is a geeal poit o C. The poit A o C has paamete t = 4. The lie l passes though A ad also passes though the focus of C. (b) Fid the gadiet of l. 1 *N35387A018*

8 physicsadmathstuto.com Jue The ectagula hypebola H has equatio The poit A o H has x-coodiate 3c. xy = c, whee c is a positive costat. blak (a) Wite dow the y-coodiate of A. (b) Show that a equatio of the omal to H at A is 3y = 7x 80c (5) The omal to H at A meets H agai at the poit B. (c) Fid, i tems of c, the coodiates of B. (5) 0 *N35387A008*

9 physicsadmathstuto.com Jauay y blak Q P O S x C Figue 1 Figue 1 shows a sketch of the paabola C with equatio y = 36x. The poit S is the focus of C. (a) Fid the coodiates of S. (b) Wite dow the equatio of the diectix of C. Figue 1 shows the poit P which lies o C, whee y > 0, ad the poit Q which lies o the diectix of C. The lie segmet QP is paallel to the x-axis. Give that the distace PS is 5, (c) wite dow the distace QP, (d) fid the coodiates of P, (3) (e) fid the aea of the tapezium OSPQ. () 14 *N35406A0143*

10 physicsadmathstuto.com Jauay The poit 6 P 6t,, t 0, lies o the ectagula hypebola H with equatio xy = 36. t blak (a) Show that a equatio fo the taget to H at P is y = 1 x + 1 t t (5) The taget to H at the poit A ad the taget to H at the poit B meet at the poit 91,. ( ) (b) Fid the coodiates of A ad B. (7) 8 *N35406A083*

11 physicsadmathstuto.com Jue The paabola C has equatio y = 48 x. blak The poit P 1t, 4t ( ) is a geeal poit o C. (a) Fid the equatio of the diectix of C. () (, ) is (b) Show that the equatio of the taget to C at P 1t 4t x t y + 1t = 0 The taget to C at the poit ( 3, 1 ) meets the diectix of C at the poit X. (c) Fid the coodiates of X. 4 *P38168A043*

12 physicsadmathstuto.com Jauay 01 blak 3. A paabola C has catesia equatio y = 16x. The poit P( 4t, 8 t) is a geeal poit o C. (a) Wite dow the coodiates of the focus F ad the equatio of the diectix of C. (3) (b) Show that the equatio of the omal to C at P is y+ tx= 8t+ 4t 3. (5) 6 *P40086A064*

13 physicsadmathstuto.com Jauay 01 blak 9. The ectagula hypebola H has catesia equatio xy = 9 3 The poits P 3p, Q 3q, 3 p ad q lie o H, whee p ± q. (a) Show that the equatio of the taget at P is x+ p y = 6 p. (b) Wite dow the equatio of the taget at Q. The taget at the poit P ad the taget at the poit Q itesect at R. (c) Fid, as sigle factios i thei simplest fom, the coodiates of R i tems of p ad q. *P40086A04*

14 physicsadmathstuto.com Jue y P C blak O S x Q Figue 1 Figue 1 shows a sketch of the paabola with equatio y = 8x. The poit lies o C, whee y > 0, ad the poit Q lies o C, whee y < 0 The lie segmet PQ is paallel to the y-axis. Give that the distace PQ is 1, (a) wite dow the y-coodiate of P, (b) fid the x-coodiate of P. () Figue 1 shows the poit S which is the focus of C. The lie l passes though the poit P ad the poit S. (c) Fid a equatio fo l i the fom ax + + c = 0, whee a, b ad c ae iteges. 10 *P40688A0103*

15 physicsadmathstuto.com Jue The ectagula hypebola H has equatio xy = c, whee c is a positive costat. blak c The poit P ct,, t, t 0 is a geeal poit o H. (a) Show that a equatio fo the taget to H at P is x + t y = ct The taget to H at the poit P meets the x-axis at the poit ad the y-axis at the poit B. Give that the aea of the tiagle OAB, whee O is the oigi, is 36, (b) fid the exact value of c, expessig you aswe i the fom k, whee k is a itege. 0 *P40688A003*

16 physicsadmathstuto.com Jauay 013 blak 7. The ectagula hypebola, H, has catesia equatio xy = 5 The poit P 5p, 5 p, ad the poit Q 5q, 5 q, whee p, q p q, ae poits o the ectagula hypebola H. (a) Show that the equatio of the taget at poit P is py+ x=10 p (b) Wite dow the equatio of the taget at poit Q. The tagets at P ad Q meet at the poit N. Give p+ q 0, 10 pq 10 (c) show that poit N has coodiates, p+ q p+ q. The lie joiig N to the oigi is pepedicula to the lie PQ. (d) Fid the value of pq. (5) 18 *P41485A0188*

17 physicsadmathstuto.com Jauay y y = 36x blak P O S N x Figue 1 Figue 1 shows a sketch of pat of the paabola with equatio y The poit P (4, 1) lies o the paabola. = 36x. (a) Fid a equatio fo the omal to the paabola at P. (5) This omal meets the x-axis at the poit N ad S is the focus of the paabola, as show i Figue 1. (b) Fid the aea of tiagle PSN. 6 *P41485A068*

18 physicsadmathstuto.com Jue The ectagula hypebola H has Catesia equatio xy = 4 The poit P t, t lies o H, whee t 0 blak (a) Show that a equatio of the omal to H at the poit P is ty t 3 x = t 4 (5) The omal to H at the poit whee t = 1 meets H agai at the poit Q. (b) Fid the coodiates of the poit Q. 8 *P43138A083*

19 physicsadmathstuto.com Jue A paabola C has equatio y = 4ax, a > 0 blak The poits P(ap, ap) ad Q(aq, aq) lie o C, whee p 0, q 0, p q. (a) Show that a equatio of the taget to the paabola at P is py x = ap (b) Wite dow the equatio of the taget at Q. The taget at P meets the taget at Q at the poit R. (c) Fid, i tems of p ad q, the coodiates of R, givig you aswes i thei simplest fom. Give that R lies o the diectix of C, (d) fid the value of pq. () 16 *P43138A0163*

20 physicsadmathstuto.com Jue 013 R

21 physicsadmathstuto.com Jue 013 R

22 Futhe Pue Mathematics FP1 Cadidates sittig FP1 may also equie those fomulae listed ude Coe Mathematics C1 ad C. Summatios = 1 = 1 3 = = ( + 1)( + 1) ( +1) Numeical solutio of equatios The Newto-Raphso iteatio fo solvig f( x ) = 0 : x + 1 f( x ) = x f ( x ) Coics Paabola Rectagula Hypebola Stadad Fom y = 4ax xy = c Paametic Fom (at, at) ct, c t Foci (a, 0) Not equied Diectices x = a Not equied Matix tasfomatios Aticlockwise otatio though θ about O: cosθ siθ siθ cosθ Reflectio i the lie cos θ si θ y = (taθ ) x : si θ cos θ I FP1, θ will be a multiple of Edexcel AS/A level Mathematics Fomulae List: Futhe Pue Mathematics FP1 Issue 1 Septembe 009

23 Coe Mathematics C1 Mesuatio Suface aea of sphee = 4π Aea of cuved suface of coe = π slat height Aithmetic seies u = a + ( 1)d S = 1 (a + l) = 1 [a + ( 1)d] 4 Edexcel AS/A level Mathematics Fomulae List: Coe Mathematics C1 Issue 1 Septembe 009

24 Edexcel AS/A level Mathematics Fomulae List: Coe Mathematics C Issue 1 Septembe Coe Mathematics C Cadidates sittig C may also equie those fomulae listed ude Coe Mathematics C1. Cosie ule a = b + c bc cos A Biomial seies 1 ) ( 1 b b a b a b a a b a = + K K ( N) whee )!!(! C = = < = + x x x x x 1, ( 1 1) ( 1) ( 1 1) ( 1 ) (1 K K K K R) Logaithms ad expoetials a x x b b a log log log = Geometic seies u = a 1 S = a 1 ) (1 S = a 1 fo < 1 Numeical itegatio The tapezium ule: b a x y d 1 h{(y 0 + y ) + (y 1 + y y 1 )}, whee a b h =

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