Download - 11X1 T12 02 parabola as a locus
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
