The Parabola As a Locus y

x

The Parabola As a Locus A point moves so that its distance

from a fixed point (focus) is

equal to its distance from a fixed

line (directrix)

y

x

y

x

0,S a

The Parabola As a Locus A point moves so that its distance

from a fixed point (focus) is

equal to its distance from a fixed

line (directrix)

y

x

0,S a

y a

The Parabola As a Locus A point moves so that its distance

from a fixed point (focus) is

equal to its distance from a fixed

line (directrix)

y

x

0,S a

y a

The Parabola As a Locus A point moves so that its distance

from a fixed point (focus) is

equal to its distance from a fixed

line (directrix)

y

x

0,S a

y a

,P x y

The Parabola As a Locus A point moves so that its distance

from a fixed point (focus) is

equal to its distance from a fixed

line (directrix)

y

x

0,S a

y a

,P x y

( , )M x a

The Parabola As a Locus A point moves so that its distance

from a fixed point (focus) is

equal to its distance from a fixed

line (directrix)

y

x

0,S a

y a

,P x y

( , )M x aPS PMd d

The Parabola As a Locus A point moves so that its distance

from a fixed point (focus) is

equal to its distance from a fixed

line (directrix)

y

x

0,S a

y a

,P x y

( , )M x aPS PMd d

2 2 2 2

0x y a x x y a

The Parabola As a Locus A point moves so that its distance

from a fixed point (focus) is

equal to its distance from a fixed

line (directrix)

y

x

0,S a

y a

,P x y

( , )M x aPS PMd d

2 2 2 2

0x y a x x y a

2 22x y a y a

The Parabola As a Locus A point moves so that its distance

from a fixed point (focus) is

equal to its distance from a fixed

line (directrix)

y

x

0,S a

y a

,P x y

( , )M x aPS PMd d

2 2 2 2

0x y a x x y a

2 22x y a y a

2 2 2 2 22 2x y ay a y ay a

The Parabola As a Locus A point moves so that its distance

from a fixed point (focus) is

equal to its distance from a fixed

line (directrix)

y

x

0,S a

y a

,P x y

( , )M x aPS PMd d

2 2 2 2

0x y a x x y a

2 22x y a y a

2 2 2 2 22 2x y ay a y ay a

2 4x ay

The Parabola As a Locus A point moves so that its distance

from a fixed point (focus) is

equal to its distance from a fixed

line (directrix)

2 4x ay

2 4x ay

vertex: 0,0

2 4x ay

vertex: 0,0

focus: 0,a

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;2a) 32x y

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

2a) 32x y

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

2a) 32x y

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

2a) 32x y

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

2a) 32x y

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

2a) 32x y

(0,0)

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

2a) 32x y

(0,0)8

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)2a) 32x y

(0,0)8

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)2a) 32x y

(0,0)8

8

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)

directrix is 8y

2a) 32x y

(0,0)8

8

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)

directrix is 8y

2a) 32x y

(0,0)8

8

2b) 4y x

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)

directrix is 8y

2a) 32x y

(0,0)8

8

2b) 4y x2 1

4x y

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)

directrix is 8y

2a) 32x y

(0,0)8

8

14

4a

2b) 4y x2 1

4x y

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)

directrix is 8y

2a) 32x y

(0,0)8

8

14

4a

1

16a

2b) 4y x2 1

4x y

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)

directrix is 8y

2a) 32x y

(0,0)8

8

14

4a

1

16a 1

focal length = unit16

2b) 4y x2 1

4x y

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)

directrix is 8y

2a) 32x y

(0,0)8

8

14

4a

1

16a 1

focal length = unit16

2b) 4y x2 1

4x y

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)

directrix is 8y

2a) 32x y

(0,0)8

8

14

4a

1

16a 1

focal length = unit16

2b) 4y x

(0,0)

2 1

4x y

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)

directrix is 8y

2a) 32x y

(0,0)8

8

14

4a

1

16a 1

focal length = unit16

2b) 4y x

(0,0)

2 1

4x y

1

16

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)

directrix is 8y

2a) 32x y

(0,0)8

8

14

4a

1

16a 1

focal length = unit16

1 focus is 0,

16

2b) 4y x

(0,0)

2 1

4x y

1

16

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)

directrix is 8y

2a) 32x y

(0,0)8

8

14

4a

1

16a 1

focal length = unit16

1 focus is 0,

16

2b) 4y x

(0,0)

2 1

4x y

1

16

1

16

2 4x ay

vertex: 0,0

focus: 0,a

directrix: y a

focal length: unitsa

e.g. (i) Find the focus, focal length and directrix;

4 32a

8a

focal length = 8 units

focus is (0,8)

directrix is 8y

2a) 32x y

(0,0)8

8

14

4a

1

16a 1

focal length = unit16

1 focus is 0,

16

1directrix is

16y

2b) 4y x

(0,0)

2 1

4x y

1

16

1

16

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

2a

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

2a 2 4 2x y

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

2a 2 4 2x y

2 8x y

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

2a 2 4 2x y

2 8x y

b) focus 3,0 , directrix 3x

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

2a 2 4 2x y

2 8x y

b) focus 3,0 , directrix 3x

3a

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

2a 2 4 2x y

2 8x y

b) focus 3,0 , directrix 3x

3a 2 4 3y x

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

2a 2 4 2x y

2 8x y

b) focus 3,0 , directrix 3x

3a 2 4 3y x

2 12y x

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

2a 2 4 2x y

2 8x y

b) focus 3,0 , directrix 3x

3a 2 4 3y x

2 12y x

Vertex NOT at the origin

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

2a 2 4 2x y

2 8x y

b) focus 3,0 , directrix 3x

3a 2 4 3y x

2 12y x

Vertex NOT at the origin

2

4x p a y q

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

2a 2 4 2x y

2 8x y

b) focus 3,0 , directrix 3x

3a 2 4 3y x

2 12y x

Vertex NOT at the origin

2

4x p a y q

vertex: ,p q

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

2a 2 4 2x y

2 8x y

b) focus 3,0 , directrix 3x

3a 2 4 3y x

2 12y x

Vertex NOT at the origin

2

4x p a y q

vertex: ,p q

focus: ,p q a

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

2a 2 4 2x y

2 8x y

b) focus 3,0 , directrix 3x

3a 2 4 3y x

2 12y x

Vertex NOT at the origin

2

4x p a y q

vertex: ,p q

focus: ,p q a

directrix: y q a

(ii) Find the equation of the parabola with;

a) focus 0, 2 , directrix 2y

2a 2 4 2x y

2 8x y

b) focus 3,0 , directrix 3x

3a 2 4 3y x

2 12y x

Vertex NOT at the origin

2

4x p a y q

vertex: ,p q

focus: ,p q a

directrix: y q a

focal length: unitsa

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

2

3 4 2 1x y

e.g. (i) Find the equation of the parabola with vertex 3,1 and

focal length 2 units

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

e.g. (i) Find the equation of the parabola with vertex 3,1 and

focal length 2 units

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 17

16 17

8

y x x

y x x

e.g. (i) Find the equation of the parabola with vertex 3,1 and

focal length 2 units

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 17

16 17

8

y x x

y x x

2

3 4 2 1x y

e.g. (i) Find the equation of the parabola with vertex 3,1 and

focal length 2 units

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 17

16 17

8

y x x

y x x

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

e.g. (i) Find the equation of the parabola with vertex 3,1 and

focal length 2 units

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 17

16 17

8

y x x

y x x

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 1

16 1

8

y x x

y x x

e.g. (i) Find the equation of the parabola with vertex 3,1 and

focal length 2 units

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 17

16 17

8

y x x

y x x

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 1

16 1

8

y x x

y x x

2

1 4 2 3y x

e.g. (i) Find the equation of the parabola with vertex 3,1 and

focal length 2 units

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 17

16 17

8

y x x

y x x

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 1

16 1

8

y x x

y x x

2

1 4 2 3y x

2

1 8 3y x 2 2 1 8 24y y x

e.g. (i) Find the equation of the parabola with vertex 3,1 and

focal length 2 units

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 17

16 17

8

y x x

y x x

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 1

16 1

8

y x x

y x x

2

1 4 2 3y x

2

1 8 3y x 2 2 1 8 24y y x

2

2

8 2 25

12 25

8

x y y

x y y

e.g. (i) Find the equation of the parabola with vertex 3,1 and

focal length 2 units

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 17

16 17

8

y x x

y x x

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 1

16 1

8

y x x

y x x

2

1 4 2 3y x

2

1 8 3y x 2 2 1 8 24y y x

2

2

8 2 25

12 25

8

x y y

x y y

2

1 4 2 3y x

e.g. (i) Find the equation of the parabola with vertex 3,1 and

focal length 2 units

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 17

16 17

8

y x x

y x x

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 1

16 1

8

y x x

y x x

2

1 4 2 3y x

2

1 8 3y x 2 2 1 8 24y y x

2

2

8 2 25

12 25

8

x y y

x y y

2

1 4 2 3y x

2

1 8 3y x 2 2 1 8 24y y x

e.g. (i) Find the equation of the parabola with vertex 3,1 and

focal length 2 units

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 17

16 17

8

y x x

y x x

2

3 4 2 1x y

2

3 8 1x y 2 6 9 8 8x x y

2

2

8 6 1

16 1

8

y x x

y x x

2

1 4 2 3y x

2

1 8 3y x 2 2 1 8 24y y x

2

2

8 2 25

12 25

8

x y y

x y y

2

1 4 2 3y x

2

1 8 3y x 2 2 1 8 24y y x

2

2

8 2 23

12 23

8

x y y

x y y

(ii) focus (2,8) and directrix y = 10

(ii) focus (2,8) and directrix y = 10

(ii) focus (2,8) and directrix y = 10

10y a

(ii) focus (2,8) and directrix y = 10

10y a

a

2,8

(ii) focus (2,8) and directrix y = 10

10y a

a

2,8

2 2a

1a

(ii) focus (2,8) and directrix y = 10

10y a

a

2,8

2 2a

1a vertex is (2,9)

(ii) focus (2,8) and directrix y = 10

10y a

a

2,8

2 2a

1a vertex is (2,9) 2

2 4 1 9x y

(ii) focus (2,8) and directrix y = 10

10y a

a

2,8

2 2a

1a vertex is (2,9) 2

2 4 1 9x y

2

2 4 9x y

(ii) focus (2,8) and directrix y = 10

10y a

a

2,8

2 2a

1a vertex is (2,9) 2

2 4 1 9x y

2

2 4 9x y 2 4 16 4 36x x y

(ii) focus (2,8) and directrix y = 10

10y a

a

2,8

2 2a

1a vertex is (2,9) 2

2 4 1 9x y

2

2 4 9x y 2 4 16 4 36x x y

2

2

4 4 20

14 20

4

y x x

y x x

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

4a

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

4a 12

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

4a 12

3a

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

4a 12

3a

focal length = 3 units

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

4a 12

3a vertex: (3,

focal length = 3 units

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

4a 12

3a vertex: (3, –1)

focal length = 3 units

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

4a 12

3a vertex: (3, –1)

focal length = 3 units

vertex = 3, 1

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

4a 12

3a vertex: (3, –1)

focal length = 3 units

vertex = 3, 1

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

4a 12

3a vertex: (3, –1)

(3, 1) focal length = 3 units

vertex = 3, 1

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

4a 12

3a vertex: (3, –1)

(3, 1)3

focal length = 3 units

vertex = 3, 1

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

4a 12

3a vertex: (3, –1)

(3, 1)3

focal length = 3 units

vertex = 3, 1

focus = 3,2

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

4a 12

3a vertex: (3, –1)

(3, 1)3

focal length = 3 units 3

vertex = 3, 1

focus = 3,2

(iii) Find the vertex, focus, focal length, directrix of 212 6 3y x x

212 6 3y x x 212 3 6y x x

2

12 3 9 3y x

2

12 12 3y x

2

12 1 3y x

4a 12

3a vertex: (3, –1)

(3, 1)3

focal length = 3 units 3

vertex = 3, 1

focus = 3,2

directrix: 4y

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