Berkari 6.19
- 1.
- 1 -
conic ections
ParabolaElipseHypebola
0,0
0,0
(0, ) ( , 0)
= − = −
= 0 = 0
- 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
- 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
- 10.
- 10 -
x
= 8
x
= −4
y
= −6
0,0
,,
- 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
- 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
- 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)
- 24.
- 24 -
,
+ = 1
= ±
3
1
= ± = 3
+ = 1
(2, ±3)
3,5= ±
= 3 ℎ = 2 = 0
(2,0)
3,5
- 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
- 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
- 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
