The Parabola As a Locus
y
x

The Parabola As a Locus
y A point moves so that its distance
from a fixed point (focus) is
equal to its distance from a fixed
line (directrix)
x

The Parabola As a Locus
y A point moves so that its distance
from a fixed point (focus) is
equal to its distance from a fixed
S  0, a  line (directrix)
x

The Parabola As a Locus
y A point moves so that its distance
from a fixed point (focus) is
equal to its distance from a fixed
S  0, a  line (directrix)
x
y  a

The Parabola As a Locus
y A point moves so that its distance
from a fixed point (focus) is
equal to its distance from a fixed
S  0, a  line (directrix)
x
y  a

The Parabola As a Locus
y A point moves so that its distance
from a fixed point (focus) is
equal to its distance from a fixed
S  0, a  P  x, y  line (directrix)
x
y  a

The Parabola As a Locus
y A point moves so that its distance
from a fixed point (focus) is
equal to its distance from a fixed
S  0, a  P  x, y  line (directrix)
x
y  a M ( x,  a )

The Parabola As a Locus
y A point moves so that its distance
from a fixed point (focus) is
equal to its distance from a fixed
S  0, a  P  x, y  line (directrix)
x
y  a M ( x,  a ) d PS  d PM

The Parabola As a Locus
y A point moves so that its distance
from a fixed point (focus) is
equal to its distance from a fixed
S  0, a  P  x, y  line (directrix)
x
y  a M ( x,  a ) d PS  d PM
 x  0   y  a    x  x    y  a 
2 2 2 2

The Parabola As a Locus
y A point moves so that its distance
from a fixed point (focus) is
equal to its distance from a fixed
S  0, a  P  x, y  line (directrix)
x
y  a M ( x,  a ) d PS  d PM
 x  0   y  a    x  x    y  a 
2 2 2 2
x2   y  a    y  a 
2 2

The Parabola As a Locus
y A point moves so that its distance
from a fixed point (focus) is
equal to its distance from a fixed
S  0, a  P  x, y  line (directrix)
x
y  a M ( x,  a ) d PS  d PM
 x  0   y  a    x  x    y  a 
2 2 2 2
x2   y  a    y  a 
2 2
x 2  y 2  2ay  a 2  y 2  2ay  a 2

The Parabola As a Locus
y A point moves so that its distance
from a fixed point (focus) is
equal to its distance from a fixed
S  0, a  P  x, y  line (directrix)
x
y  a M ( x,  a ) d PS  d PM
 x  0   y  a    x  x    y  a 
2 2 2 2
x2   y  a    y  a 
2 2
x 2  y 2  2ay  a 2  y 2  2ay  a 2
x 2  4ay

x 2  4ay

x 2  4ay
vertex:  0,0 

x 2  4ay
vertex:  0,0 
focus:  0, a 

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y
4a  32

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y
4a  32
a 8

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y
4a  32
a 8
focal length = 8 units

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y
4a  32
a 8
focal length = 8 units

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y
4a  32
a 8 (0,0)
focal length = 8 units

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y
4a  32
8
a 8 (0,0)
focal length = 8 units

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 (0,0)
focal length = 8 units

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 (0,0) 8
focal length = 8 units

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 directrix is y  8 (0,0) 8
focal length = 8 units

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 directrix is y  8 (0,0) 8
focal length = 8 units
b) y  4 x 2

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 directrix is y  8 (0,0) 8
focal length = 8 units
1
b) y  4 x 2  x 2  y
4

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 directrix is y  8 (0,0) 8
focal length = 8 units
1
b) y  4 x 2  x 2  y
1 4
4a 
4

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 directrix is y  8 (0,0) 8
focal length = 8 units
1
b) y  4 x 2  x 2  y
1 4
4a 
4
1
a
16

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 directrix is y  8 (0,0) 8
focal length = 8 units
1
b) y  4 x 2  x 2  y
1 4
4a 
4
1
a focal length =
1
unit
16 16

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 directrix is y  8 (0,0) 8
focal length = 8 units
1
b) y  4 x 2  x 2  y
1 4
4a 
4
1
a focal length =
1
unit
16 16

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 directrix is y  8 (0,0) 8
focal length = 8 units
1
b) y  4 x 2  x 2  y
1 4
4a 
4
1 (0,0)
a focal length =
1
unit
16 16

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 directrix is y  8 (0,0) 8
focal length = 8 units
1
b) y  4 x 2  x 2  y
1 4
4a  1
4 16
1 (0,0)
a focal length =
1
unit
16 16

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 directrix is y  8 (0,0) 8
focal length = 8 units
 0, 1 
focus is 
b) y  4 x  x  y2 1 
 16 
2
1 4
4a  1
4 16
1 (0,0)
a focal length =
1
unit
16 16

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 directrix is y  8 (0,0) 8
focal length = 8 units
 0, 1 
focus is 
b) y  4 x  x  y2 1 
 16 
2
1 4
4a  1
4 16
1 (0,0)
a
1
1 16
16 focal length = unit
16

x 2  4ay
vertex:  0,0 
focus:  0, a 
directrix: y  a
focal length: a units
e.g. (i) Find the focus, focal length and directrix;
a) x 2  32 y focus is (0,8)
4a  32
8
a 8 directrix is y  8 (0,0) 8
focal length = 8 units
 0, 1 
focus is 
b) y  4 x  x  y2 1 
 16 
2
1 4
4a  1
4 directrix is y   1
1 16 16
a 1 (0,0) 1
16
16 focal length = unit
16

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2
a  2

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2
a  2 x 2  4  2  y

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2
a  2 x 2  4  2  y
x 2  8 y

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2 b) focus  3,0  , directrix x  3
a  2 x 2  4  2  y
x 2  8 y

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2 b) focus  3,0  , directrix x  3
a  2 x 2  4  2  y a3
x 2  8 y

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2 b) focus  3,0  , directrix x  3
a  2 x 2  4  2  y a3 y 2  4  3 x
x 2  8 y

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2 b) focus  3,0  , directrix x  3
a  2 x 2  4  2  y a3 y 2  4  3 x
x 2  8 y y 2  12 x

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2 b) focus  3,0  , directrix x  3
a  2 x 2  4  2  y a3 y 2  4  3 x
x 2  8 y y 2  12 x
Vertex NOT at the origin

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2 b) focus  3,0  , directrix x  3
a  2 x 2  4  2  y a3 y 2  4  3 x
x 2  8 y y 2  12 x
Vertex NOT at the origin
 x  p   4a  y  q 
2

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2 b) focus  3,0  , directrix x  3
a  2 x 2  4  2  y a3 y 2  4  3 x
x 2  8 y y 2  12 x
Vertex NOT at the origin
 x  p   4a  y  q 
2
vertex:  p, q 

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2 b) focus  3,0  , directrix x  3
a  2 x 2  4  2  y a3 y 2  4  3 x
x 2  8 y y 2  12 x
Vertex NOT at the origin
 x  p   4a  y  q 
2
vertex:  p, q 
focus:  p, q  a 

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2 b) focus  3,0  , directrix x  3
a  2 x 2  4  2  y a3 y 2  4  3 x
x 2  8 y y 2  12 x
Vertex NOT at the origin
 x  p   4a  y  q 
2
vertex:  p, q 
focus:  p, q  a 
directrix: y  q  a

(ii) Find the equation of the parabola with;
a) focus  0, 2  , directrix y  2 b) focus  3,0  , directrix x  3
a  2 x 2  4  2  y a3 y 2  4  3 x
x 2  8 y y 2  12 x
Vertex NOT at the origin
 x  p   4a  y  q 
2
vertex:  p, q 
focus:  p, q  a 
directrix: y  q  a
focal length: a units

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2
x2  6x  9  8 y  8

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2
x2  6x  9  8 y  8
8 y  x 2  6 x  17
y   x  6 x  17 
1 2
8

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2
x2  6x  9  8 y  8
8 y  x 2  6 x  17
y   x  6 x  17 
1 2
8
(ii) focus (2,8) and directrix y = 10

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2
x2  6x  9  8 y  8
8 y  x 2  6 x  17
y   x  6 x  17 
1 2
8
(ii) focus (2,8) and directrix y = 10

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2
x2  6x  9  8 y  8
8 y  x 2  6 x  17
y   x  6 x  17 
1 2
8
(ii) focus (2,8) and directrix y = 10
a y  10

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2
x2  6x  9  8 y  8
8 y  x 2  6 x  17
y   x  6 x  17 
1 2
8
(ii) focus (2,8) and directrix y = 10
a y  10
a
 2,8

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2
x2  6x  9  8 y  8
8 y  x 2  6 x  17
y   x  6 x  17 
1 2
8
(ii) focus (2,8) and directrix y = 10
a y  10
a
 2,8
2a  2

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2
x2  6x  9  8 y  8
8 y  x 2  6 x  17
y   x  6 x  17 
1 2
8
(ii) focus (2,8) and directrix y = 10
a y  10
a
 2,8
2a  2
a 1

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2
x2  6x  9  8 y  8
8 y  x 2  6 x  17
y   x  6 x  17 
1 2
8
(ii) focus (2,8) and directrix y = 10
a y  10
a
 2,8
2a  2
a  1 vertex is (2,9)

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2
x2  6x  9  8 y  8
8 y  x 2  6 x  17
y   x  6 x  17 
1 2
8
(ii) focus (2,8) and directrix y = 10
 x  2   4 1 y  9 
2
a y  10
a
 2,8
2a  2
a  1 vertex is (2,9)

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2
x2  6x  9  8 y  8
8 y  x 2  6 x  17
y   x  6 x  17 
1 2
8
(ii) focus (2,8) and directrix y = 10
 x  2   4 1 y  9 
2
a y  10
 x  2   4  y  9 
2
a
 2,8
2a  2
a  1 vertex is (2,9)

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2
x2  6x  9  8 y  8
8 y  x 2  6 x  17
y   x  6 x  17 
1 2
8
(ii) focus (2,8) and directrix y = 10
 x  2   4 1 y  9 
2
a y  10
 x  2   4  y  9 
2
a
 2,8 x 2  4 x  16  4 y  36
2a  2
a  1 vertex is (2,9)

e.g. (i) Find the equation of the parabola with vertex  3,1 and
focal length 2 units
 x  3  4  2  y  1
2
 x  3  8  y  1
2
x2  6x  9  8 y  8
8 y  x 2  6 x  17
y   x  6 x  17 
1 2
8
(ii) focus (2,8) and directrix y = 10
 x  2   4 1 y  9 
2
a y  10
 x  2   4  y  9 
2
a
 2,8 x 2  4 x  16  4 y  36
4 y   x 2  4 x  20
2a  2
y    x  4 x  20 
1 2
a  1 vertex is (2,9) 4

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2
4a

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2
4a  12

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2
4a  12
a3

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2
4a  12
a3
 focal length = 3 units

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2
4a  12
a3 vertex: (3,
 focal length = 3 units

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2
4a  12
a3 vertex: (3, –1)
 focal length = 3 units

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2
4a  12
a3 vertex: (3, –1)
 focal length = 3 units
vertex =  3, 1

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2
4a  12
a3 vertex: (3, –1)
 focal length = 3 units
vertex =  3, 1

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2
4a  12
a3 vertex: (3, –1)
(3, 1)
 focal length = 3 units
vertex =  3, 1

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2
4a  12
a3 vertex: (3, –1)
3
(3, 1)
 focal length = 3 units
vertex =  3, 1

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2
4a  12
a3 vertex: (3, –1)
3
(3, 1)
 focal length = 3 units
vertex =  3, 1
focus =  3, 2 

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2
4a  12
a3 vertex: (3, –1)
3
(3, 1)
 focal length = 3 units 3
vertex =  3, 1
focus =  3, 2 

(iii) Find the vertex, focus, focal length, directrix of 12 y  x 2  6 x  3
12 y  x 2  6 x  3
12 y  3  x 2  6 x
12 y  3  9   x  3
2
12 y  12   x  3
2
12  y  1   x  3
2
4a  12
a3 vertex: (3, –1)
3
(3, 1)
 focal length = 3 units 3
vertex =  3, 1
focus =  3, 2 
directrix: y  4

Exercise 9B; 1,2 try at home
4 (use definition)
6ace etc, 7ac, 8ace, 9ace, 10ac, 11bd, 12a
Exercise 9C; 3 to 8 ace etc, 10ac, 11ace, 12