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Berkari 6.19
- 1. - 1 - conicections ParabolaElipseHypebola 0,0 0,0 (0, ) ( , 0) = − = − = 0 = 0
- 2. - 2 - = 4 0,0 0,0 = 12 = 4 4 = 12 = 3 F(3,0) F(P, 0) = − = −3
- 3. - 3 - =−6 = 4 4 = −6 ℎ,
- 4. - 4 - (− ) = 4 ( − ℎ)( − ℎ) = 4 ( − ) = 4 = 4 −ℎ− (ℎ, + ) ( , 0) (ℎ, + ) (0, ) = ℎ − = − = = 0 = − = − = ℎ = 0 ( + 3) + ( − 2) = 0 ( − 2) = −( − 3) ( − ) = 4P( − ℎ) 4 = −1 = −1 4 h = −3 , K = 2
- 5. - 5 - −= 0 = (ℎ + , ) (−3 − 1 4 , 2) = ( −13 4 , 2) = ℎ − = −3 + 1 4 = −11 4 4 = −8 = −2 ℎ = −2 K = −2 2,-2 h, k + 4y + 8 − 12 = 0 + 4y + 0 = −8 + 12 + 0 + 4y + 4 = −8 + 12 + 4 ( − 2) = −8( − 2) ( − ) = 4P( − ℎ) (ℎ + , ) ( , ) (0, −2) = ℎ − = 2 + 2 = 4 − = 0 = = −2
- 6. - 6 - 4 = −2 = −2 4 = −1 2 ℎ = 1 K = −1 2 + + 2y − 1 = 0 + 2y + 1 = −2 + 1 + 1 ( + 1) = −2( + 1) ( − ) = −2( − ℎ) − = 0 = −1 (ℎ + , ) (1 − 1 2 , −1) = ( 1 2 , −1) = ℎ − = 1 + 1 2 = 3 2
- 7. - 7 - (ℎ, ) (3,2 ) (1,2 ) = 2 ℎ = 3 K = 2 (ℎ + , ) = (1,2) ℎ + = 1 , = 2 3 + = 1 ⇒ = −2 ( − ) = 4P( − ℎ) ( − 2) = 4(−2)( − 3) ( − 2) = −8( − 3) = −2 ℎ = 0 K = 4 (0,4 ) y ( − ℎ) = 4 ( − ) −2 = 4 − ⇒ = 6 = − ⇒ = −2
- 8. - 8 - = −2 (2,2 ) = ℎ − −2 = ℎ − .. . . . .1 (ℎ + , ) = (2,2) ℎ + = 2. . . .2 , = 2 −2 = ℎ − 2 = ℎ + 0 = 2ℎ ℎ = 0 h −2 = 0 − = 2
- 9. - 9 - (4,0)(0,0)(2,4 ) 0,0 = 4 16 ⇒ = −1 4 1-6197 y= 2
- 10. - 10 - x =8 x = −4 y = −6 0,0 ,,
- 11.
- 12. - 12 - , , = − > , > : 2 : 2 5 + 7 = 70 ÷ 70 5 70 + 7 70 = 70 70
- 13. - 13 - 14 + 10 =1 = 10 = √10 = 14 = √14 = − = 14 − 10 = 4 = 2 *0,0 = 2 ( , 0), (− , 0) (2,0), (−2,0) ( , 0), (− , 0) √14, 0 , −√14, 0 2 = 2 = 2√14 2 = 2 = 2√10
- 14. - 14 - (0,0) =2 (2,0) = 3 (3,0) = − 4 = 9 − = 5 9 + 5 = 1 (0, ±5)
- 15. - 15 - 14= =− 25 = 49 − = 24 ℎ, 0,0
- 16. - 16 - (ℎ± , ) (ℎ, ± ) 9 + 25 − 36 + 150 + 36 = 0 9 + 25 − 36 + 150 + 36 = −36 9( + 4 + 4) + 25( + 6 + 9) = −36 + 36 + 225 9( − 2) + 25( + 3) = 225 ÷ 225
- 17. - 17 - (− 2) 25 + ( + 3) 9 = 1 = 25 = 9 = − = 25 − 9 = 16= 5 = 3 *2,-3 = 4 (ℎ + , ), (ℎ − , ) (6, −3), (−2,−3) (ℎ + , ), (ℎ − , ) (7, −3), (−3, −3) (3,1) (3,9)
- 18. - 18 - (ℎ,+ ), (ℎ, − ) (3,9), (3,1) (1,2)(3,2) (1,6) (1,2)ℎ = 1 = 2
- 19. - 19 - (− 1) + ( − 2) = 1 1,6 (1 − 1) + (6 − 2) = 1 ⇒ 0 + 16 = 1 = 16 3,2 (3 − 1) + (2 − 2) 16 = 1 ⇒ 4 + 0 = 1 = 16 ( − 1) 4 + ( − 2) 16 = 1
- 20. - 20 - 0,0 , , (± , 0) (0, ± )
- 21. - 21 - ℎ, = = = (ℎ ± , ) (0, ± )
- 22. - 22 - 9− − 36 − 6 + 18 = 0 9 − 36 − − 6 = −18 9 × 4 = 36 − 9 9( − 4 + 4) − ( + 6 + 9) = −18 + 36 − 9 9( − 2) − ( + 3) = 9 ÷ 9 ( − 2) 1 + ( + 3) 9 = 1 = 1 = 9 ℎ = 2 = −3= 5 = 3 = + 2,-3 = √10 (ℎ + , ), (ℎ − , ) (2 + √10, −3), (2 − √10, −3) (ℎ + , ), (ℎ − , ) (3, −3), (1, −3)
- 23. - 23 - (±1,0) 3ℎ,,= ± = 1 ℎ = 0 = 0 (0,0)
- 24. - 24 - , + = 1 = ± 3 1 = ± = 3 + = 1 (2, ±3) 3,5= ± = 3 ℎ = 2 = 0 (2,0) 3,5
- 25.
- 26. - 26 - (+ 2) = 8( − 1) (+1, −2) = 8 (0,0) ( , 0) = 8 = 4 4 = 8 = 2 = 2 ( + 2) = 8( − 1) (1, −2) (ℎ + , ) ℎ = 1 = −2
- 27.
- 28. - 28 - ℎ, −ℎ = ± d p = = 1 < 1 > 1
- 29. - 29 - (0,±6.5) 0.75 = ( ) 0 (0, ±8.7) 16 − 9 = 144 9 + 16 = 1 = 16 = 9 = 4 = 3 = + =
- 30. - 30 - = 5 3 =| − 4| (0,0)= 4(3,0) ( , )(3,0) = 4 = 4 = = 4 = c = 3 4 = 3 4 = √3 2 ( − 3) + = 3 4 ( − 8 + 16) − 6 + 9 + = 3 4 ( − 8 + 16) 4 − 24 + 36 + 4 = 3 − 24 + 48 + 4 = 48 − 36 + 4 = 12 ÷ 12 + =1 = 2 (1, −3) =
- 31. - 31 - = 9 6 =C = 9 = 6 =(1, −3) = 2= | − 4| ( , ) ( − 1) + + 6 + 9 = 9 4 ( − 4 + 4) 4( − 1) + 4 + 24 + 36 = 9 − 36 + 36 4( − 1) − 5 + 60 = 0 4( − 1) − 5( + 12 + 36) = −180 4( − 1) − 5( − 6) = −180 ÷ −180 ( − 6) 36 − ( − 1) 45 = 1 (0,0) = 4 5 = 4
- 32. - 32 - =5 = + = | − | = | − | (1,4)ℎ = 1 = 4 2 = |7 − 1| 2 = 6 = 3 2 = | − 1 − 3| 2 = 4 = 2 ( − ℎ) b + ( − K) = 1 ( ) + ( ) = 1 = + = 9 − 4 = 5 = √5 = = √5 3
- 33. - 33 - (ℎ,+ ), (ℎ, − ) 1,4 + √5 , 1,4 − √5 (1,4) (0, ±1) = 3 − = 1 3 = 3 = + 9 = 1 + = 8 1 − 8 = 1 4 + + + + = 0 , ,, , −1,2 4(−1) + (2) + (−1) + (2) + = 0 4 + 4 − + 2 + = 0 2 − + = −8 … … … … 1
- 34. - 34 - 0,0 4(0)+ (0) + (0) + (0) + = 0 = 0 0,0̀ = 0 8 + 2 ̀ + + ̀ + 0 = 0 8(0) + 2(0)(0) + + (0) + 0 = 0 = 0 C, a 4 + + + 4 = 0 4 + − 4 + 4 = 0 + 4 4 + ( − 2) = 4 ÷ 4 1 + ( − 2) 4 = 1 = 1 = 1 = + = 4 − 1 = 3 = √3 = 4 = 2 = = √3 2
- 35. - 35 - 6= (− ℎ) + ( − K) = 1 (ℎ + , ), (ℎ − , ) (10,3), (2,2) ℎ ℎ + = 10 . . . . . . . .1 ℎ − = 2 . . . . . . . . .2 2ℎ = 12 ℎ = 6 = + 16 = 9 + = 7
- 36. - 36 - 6= (10,2), (2,2) ( − 10) + ( − 2) − ( − 2) + ( − 2) = ±6 ( − 10) + ( − 2) = ±6 + ( − 2) + ( − 2) = 1 4 = 1 4 − 9 = 36 ÷ 36 9 − 4 = 1