The Parabola As a Locusy

x

The Parabola As a LocusA 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 LocusA 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 LocusA 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 LocusA 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 LocusA 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 LocusA 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 LocusA 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 20x y a x x y a

The Parabola As a LocusA 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 20x y a x x y a

2 22x y a y a

The Parabola As a LocusA 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 20x 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 LocusA 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 20x 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 LocusA 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 x 2 14

x 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

144

a

2b) 4y x 2 14

x 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

144

a

116

a

2b) 4y x 2 14

x 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

144

a

116

a 1focal length = unit16

2b) 4y x 2 14

x 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

144

a

116

a 1focal length = unit16

2b) 4y x 2 14

x 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

144

a

116

a 1focal length = unit16

2b) 4y x

(0,0)

2 14

x 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

144

a

116

a 1focal length = unit16

2b) 4y x

(0,0)

2 14

x y

116

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

144

a

116

a 1focal length = unit16

1focus is 0,16

2b) 4y x

(0,0)

2 14

x y

116

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

144

a

116

a 1focal length = unit16

1focus is 0,16

2b) 4y x

(0,0)

2 14

x y

1161

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

144

a

116

a 1focal length = unit16

1focus is 0,16

1directrix is 16

y

2b) 4y x

(0,0)

2 14

x y

1161

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

23 4 2 1x y

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

23 4 2 1x y 23 8 1x y

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

23 4 2 1x y 23 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

23 4 2 1x y 23 8 1x y

2 6 9 8 8x x y

2

2

8 6 171 6 178

y x x

y x x

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

23 4 2 1x y 23 8 1x y

2 6 9 8 8x x y

2

2

8 6 171 6 178

y x x

y x x

(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

23 4 2 1x y 23 8 1x y

2 6 9 8 8x x y

2

2

8 6 171 6 178

y x x

y x x

(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

23 4 2 1x y 23 8 1x y

2 6 9 8 8x x y

2

2

8 6 171 6 178

y x x

y x x

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

10y a

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

23 4 2 1x y 23 8 1x y

2 6 9 8 8x x y

2

2

8 6 171 6 178

y x x

y x x

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

10y a

a 2,8

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

23 4 2 1x y 23 8 1x y

2 6 9 8 8x x y

2

2

8 6 171 6 178

y x x

y x x

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

10y a

a 2,8

2 2a

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

23 4 2 1x y 23 8 1x y

2 6 9 8 8x x y

2

2

8 6 171 6 178

y x x

y x x

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

10y a

a 2,8

2 2a 1a

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

23 4 2 1x y 23 8 1x y

2 6 9 8 8x x y

2

2

8 6 171 6 178

y x x

y x x

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

10y a

a 2,8

2 2a 1a vertex is (2,9)

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

23 4 2 1x y 23 8 1x y

2 6 9 8 8x x y

2

2

8 6 171 6 178

y x x

y x x

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

10y a

a 2,8

2 2a 1a vertex is (2,9)

22 4 1 9x y

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

23 4 2 1x y 23 8 1x y

2 6 9 8 8x x y

2

2

8 6 171 6 178

y x x

y x x

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

10y a

a 2,8

2 2a 1a vertex is (2,9)

22 4 1 9x y 22 4 9x y

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

23 4 2 1x y 23 8 1x y

2 6 9 8 8x x y

2

2

8 6 171 6 178

y x x

y x x

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

10y a

a 2,8

2 2a 1a vertex is (2,9)

22 4 1 9x y 22 4 9x y

2 4 16 4 36x x y

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

23 4 2 1x y 23 8 1x y

2 6 9 8 8x x y

2

2

8 6 171 6 178

y x x

y x x

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

10y a

a 2,8

2 2a 1a vertex is (2,9)

22 4 1 9x y 22 4 9x y

2 4 16 4 36x x y

2

2

4 4 201 4 204

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

212 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

212 3 9 3y x

212 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

212 3 9 3y x

212 12 3y x

212 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

212 3 9 3y x

212 12 3y x

212 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

212 3 9 3y x

212 12 3y x

212 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

212 3 9 3y x

212 12 3y x

212 1 3y x

4a 123a

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

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

212 3 9 3y x

212 12 3y x

212 1 3y x

4a 123a

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

212 3 9 3y x

212 12 3y x

212 1 3y x

4a 123a 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

212 3 9 3y x

212 12 3y x

212 1 3y x

4a 123a 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

212 3 9 3y x

212 12 3y x

212 1 3y x

4a 123a 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

212 3 9 3y x

212 12 3y x

212 1 3y x

4a 123a 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

212 3 9 3y x

212 12 3y x

212 1 3y x

4a 123a 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

212 3 9 3y x

212 12 3y x

212 1 3y x

4a 123a 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

212 3 9 3y x

212 12 3y x

212 1 3y x

4a 123a 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

212 3 9 3y x

212 12 3y x

212 1 3y x

4a 123a 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

212 3 9 3y x

212 12 3y x

212 1 3y x

4a 123a 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 home4 (use definition)

6ace etc, 7ac, 8ace, 9ace, 10ac, 11bd, 12a

Exercise 9C; 3 to 8 ace etc, 10ac, 11ace, 12