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# X2 t01 03 argand diagram (2013)

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• 1. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 2 1 -4 -3 -2 -1 -1 -2 -3 1 2 3 4 x (real axis)
• 2. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 A=2 2 1 A -4 -3 -2 -1 1 2 3 4 x (real axis) -1 -2 -3
• 3. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 A=2 2 B = -3i 1 A -4 -3 -2 -1 1 2 3 4 x (real axis) -1 -2 -3 B
• 4. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 A=2 2 B = -3i C = -2 + i C 1 A -4 -3 -2 -1 1 2 3 4 x (real axis) -1 -2 -3 B
• 5. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 A=2 2 B = -3i C = -2 + i C 1 A D=4-i -4 -3 -2 -1 1 2 3 4 x (real axis) -1 D -2 -3 B
• 6. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 A=2 NOTE: Conjugates 2 B = -3i are reflected in the E real (x) axis C = -2 + i C 1 A D=4-i -4 -3 -2 -1 1 2 3 4 x (real axis) E=4+i -1 D -2 -3 B
• 7. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4 3 A=2 NOTE: Conjugates 2 B = -3i are reflected in the E real (x) axis C = -2 + i F C 1 A D=4-i -4 -3 -2 -1 1 2 3 4 x (real axis) E=4+i -1 D F = -(4 - i) -2 = - 4 + i NOTE: Opposites are -3 B rotated 180&#xB0;
• 8. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4G NOTE: Multiply 3 A=2 NOTE: Conjugates by i results in 2 B = -3i 90&#xB0; rotation are reflected in the E real (x) axis C = -2 + i F C 1 A D=4-i -4 -3 -2 -1 1 2 3 4 x (real axis) E=4+i -1 D F = -(4 - i) -2 = - 4 + i NOTE: G = i(4 - i) Opposites are -3 B rotated 180&#xB0; = 1+4i
• 9. The Argand Diagram Complex numbers can be represented geometrically on an Argand y (imaginary axis) Diagram. 4G NOTE: Multiply 3 A=2 NOTE: Conjugates by i results in 2 B = -3i 90&#xB0; rotation are reflected in the E real (x) axis C = -2 + i F C 1 A D=4-i -4 -3 -2 -1 1 2 3 4 x (real axis) E=4+i -1 D F = -(4 - i) -2 = - 4 + i NOTE: G = i(4 - i) Opposites are -3 B rotated 180&#xB0; = 1+4i Every complex number can be represented by a unique point on the Argand Diagram.
• 10. Locus in Terms of Complex Numbers Horizontal and Vertical Lines
• 11. Locus in Terms of Complex Numbers Horizontal and Vertical Lines y c x Im&#xF028; z &#xF029; &#xF03D; c
• 12. Locus in Terms of Complex Numbers Horizontal and Vertical Lines y c y x Im&#xF028; z &#xF029; &#xF03D; c k x Re&#xF028; z &#xF029; &#xF03D; k
• 13. y R Circles zz &#xF03D; R 2 &#xF02D;R R &#xF02D;R x
• 14. y R Circles zz &#xF03D; R 2 &#xF02D;R R x &#xF02D;R e.g.Find and describe the locus of points in the Argand diagram; (i) &#xF028; z &#xF02B; 4 &#xF02B; i &#xF029;&#xF028; z &#xF02B; 4 &#xF02D; i &#xF029; &#xF03D; 49
• 15. y R Circles zz &#xF03D; R 2 &#xF02D;R R x &#xF02D;R e.g.Find and describe the locus of points in the Argand diagram; (i) &#xF028; z &#xF02B; 4 &#xF02B; i &#xF029;&#xF028; z &#xF02B; 4 &#xF02D; i &#xF029; &#xF03D; 49 z z &#xF02B; (4 &#xF02B; i ) z &#xF02B; (4 &#xF02D; i ) z &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49
• 16. y R Circles zz &#xF03D; R 2 &#xF02D;R R x &#xF02D;R e.g.Find and describe the locus of points in the Argand diagram; (i) &#xF028; z &#xF02B; 4 &#xF02B; i &#xF029;&#xF028; z &#xF02B; 4 &#xF02D; i &#xF029; &#xF03D; 49 z z &#xF02B; (4 &#xF02B; i ) z &#xF02B; (4 &#xF02D; i ) z &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 z z &#xF02B; 4( z &#xF02B; z ) &#xF02B; iz &#xF02D; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 z z &#xF02B; 4( z &#xF02B; z ) &#xF02D; iz &#xF02D; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 &#xF028; &#xF029; z z &#xF02B; 4( z &#xF02B; z ) &#xF02D; iz &#xF02B; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 x 2 &#xF02B; y 2 &#xF02B; 8 x &#xF02B; 2 y &#xF02B; 16 &#xF02B; 1 &#xF03D; 49
• 17. y R Circles zz &#xF03D; R 2 &#xF02D;R R x &#xF02D;R e.g.Find and describe the locus of points in the Argand diagram; (i) &#xF028; z &#xF02B; 4 &#xF02B; i &#xF029;&#xF028; z &#xF02B; 4 &#xF02D; i &#xF029; &#xF03D; 49 z z &#xF02B; (4 &#xF02B; i ) z &#xF02B; (4 &#xF02D; i ) z &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 z z &#xF02B; 4( z &#xF02B; z ) &#xF02B; iz &#xF02D; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 z z &#xF02B; 4( z &#xF02B; z ) &#xF02D; iz &#xF02D; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 &#xF028; &#xF029; z z &#xF02B; 4( z &#xF02B; z ) &#xF02D; iz &#xF02B; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 x 2 &#xF02B; y 2 &#xF02B; 8 x &#xF02B; 2 y &#xF02B; 16 &#xF02B; 1 &#xF03D; 49 x 2 &#xF02B; 8 x &#xF02B; 16 &#xF02B; y 2 &#xF02B; 2 y &#xF02B; 1 &#xF03D; 49
• 18. y R Circles zz &#xF03D; R 2 &#xF02D;R R x &#xF02D;R e.g.Find and describe the locus of points in the Argand diagram; (i) &#xF028; z &#xF02B; 4 &#xF02B; i &#xF029;&#xF028; z &#xF02B; 4 &#xF02D; i &#xF029; &#xF03D; 49 z z &#xF02B; (4 &#xF02B; i ) z &#xF02B; (4 &#xF02D; i ) z &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 x 2 &#xF02B; 8 x &#xF02B; 16 &#xF02B; y 2 &#xF02B; 2 y &#xF02B; 1 &#xF03D; 49 z z &#xF02B; 4( z &#xF02B; z ) &#xF02B; iz &#xF02D; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 &#xF028; x &#xF02B; 4 &#xF029; &#xF02B; &#xF028; y &#xF02B; 1&#xF029; z z &#xF02B; 4( z &#xF02B; z ) &#xF02D; iz &#xF02D; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 &#xF028; &#xF029; z z &#xF02B; 4( z &#xF02B; z ) &#xF02D; iz &#xF02B; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 x 2 &#xF02B; y 2 &#xF02B; 8 x &#xF02B; 2 y &#xF02B; 16 &#xF02B; 1 &#xF03D; 49 2 2 &#xF03D; 49
• 19. y R Circles zz &#xF03D; R 2 &#xF02D;R R x &#xF02D;R e.g.Find and describe the locus of points in the Argand diagram; (i) &#xF028; z &#xF02B; 4 &#xF02B; i &#xF029;&#xF028; z &#xF02B; 4 &#xF02D; i &#xF029; &#xF03D; 49 z z &#xF02B; (4 &#xF02B; i ) z &#xF02B; (4 &#xF02D; i ) z &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 x 2 &#xF02B; 8 x &#xF02B; 16 &#xF02B; y 2 &#xF02B; 2 y &#xF02B; 1 &#xF03D; 49 z z &#xF02B; 4( z &#xF02B; z ) &#xF02B; iz &#xF02D; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 &#xF028; x &#xF02B; 4 &#xF029; &#xF02B; &#xF028; y &#xF02B; 1&#xF029; z z &#xF02B; 4( z &#xF02B; z ) &#xF02D; iz &#xF02D; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 &#xF028; &#xF029; z z &#xF02B; 4( z &#xF02B; z ) &#xF02D; iz &#xF02B; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 x 2 &#xF02B; y 2 &#xF02B; 8 x &#xF02B; 2 y &#xF02B; 16 &#xF02B; 1 &#xF03D; 49 2 2 Locus is a circle centre : &#xF028;&#xF02D; 4 , &#xF02D; 1&#xF029; radius : 7 units &#xF03D; 49
• 20. y R Circles &#xF028; z &#xF02D; &#xF077; &#xF029;&#xF028; z &#xF02D; &#xF077; &#xF029; &#xF03D; R 2 zz &#xF03D; R 2 &#xF02D;R R x Locus is a circle centre &#x3C9; radius R &#xF02D;R e.g.Find and describe the locus of points in the Argand diagram; (i) &#xF028; z &#xF02B; 4 &#xF02B; i &#xF029;&#xF028; z &#xF02B; 4 &#xF02D; i &#xF029; &#xF03D; 49 z z &#xF02B; (4 &#xF02B; i ) z &#xF02B; (4 &#xF02D; i ) z &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 x 2 &#xF02B; 8 x &#xF02B; 16 &#xF02B; y 2 &#xF02B; 2 y &#xF02B; 1 &#xF03D; 49 z z &#xF02B; 4( z &#xF02B; z ) &#xF02B; iz &#xF02D; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 &#xF028; x &#xF02B; 4 &#xF029; &#xF02B; &#xF028; y &#xF02B; 1&#xF029; z z &#xF02B; 4( z &#xF02B; z ) &#xF02D; iz &#xF02D; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 &#xF028; &#xF029; z z &#xF02B; 4( z &#xF02B; z ) &#xF02D; iz &#xF02B; iz &#xF02B; (4 &#xF02B; i )(4 &#xF02D; i ) &#xF03D; 49 x 2 &#xF02B; y 2 &#xF02B; 8 x &#xF02B; 2 y &#xF02B; 16 &#xF02B; 1 &#xF03D; 49 2 2 Locus is a circle centre : &#xF028;&#xF02D; 4 , &#xF02D; 1&#xF029; radius : 7 units &#xF03D; 49
• 21. 1 1 (ii ) &#xF02B; &#xF03D; 1 z z
• 22. 1 1 (ii ) &#xF02B; &#xF03D; 1 z z z &#xF02B; z &#xF03D; zz
• 23. 1 1 (ii ) &#xF02B; &#xF03D; 1 z z z &#xF02B; z &#xF03D; zz 2x &#xF03D; x 2 &#xF02B; y 2
• 24. 1 1 (ii ) &#xF02B; &#xF03D; 1 z z z &#xF02B; z &#xF03D; zz 2x &#xF03D; x 2 &#xF02B; y 2 x2 &#xF02D; 2 x &#xF02B; y 2 &#xF03D; 0 ( x &#xF02D; 1) 2 &#xF02B; y 2 &#xF03D; 1
• 25. 1 1 (ii ) &#xF02B; &#xF03D; 1 z z z &#xF02B; z &#xF03D; zz 2x &#xF03D; x 2 &#xF02B; y 2 x2 &#xF02D; 2 x &#xF02B; y 2 &#xF03D; 0 ( x &#xF02D; 1) 2 &#xF02B; y 2 &#xF03D; 1 Locus is a circle centre: &#xF028;1,0 &#xF029; radius: 1 unit
• 26. 1 1 (ii ) &#xF02B; &#xF03D; 1 z z z &#xF02B; z &#xF03D; zz Locus is a circle centre: &#xF028;1,0 &#xF029; radius: 1 unit 2x &#xF03D; x 2 &#xF02B; y 2 x2 &#xF02D; 2 x &#xF02B; y 2 &#xF03D; 0 y ( x &#xF02D; 1) 2 &#xF02B; y 2 &#xF03D; 1 1 2 x
• 27. 1 1 (ii ) &#xF02B; &#xF03D; 1 z z z &#xF02B; z &#xF03D; zz Locus is a circle centre: &#xF028;1,0 &#xF029; radius: 1 unit 2x &#xF03D; x 2 &#xF02B; y 2 x2 &#xF02D; 2 x &#xF02B; y 2 &#xF03D; 0 y ( x &#xF02D; 1) 2 &#xF02B; y 2 &#xF03D; 1 excluding the point (0,0) 1 2 x
• 28. 1 1 (ii ) &#xF02B; &#xF03D; 1 z z z &#xF02B; z &#xF03D; zz Locus is a circle centre: &#xF028;1,0 &#xF029; radius: 1 unit 2x &#xF03D; x 2 &#xF02B; y 2 x2 &#xF02D; 2 x &#xF02B; y 2 &#xF03D; 0 y ( x &#xF02D; 1) 2 &#xF02B; y 2 &#xF03D; 1 excluding the point (0,0) 1 2 x Cambridge: Exercise 1C; 1 ace, 2 bdf, 3, 4 ace, 5 bdfh, 6, 10 to 15