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10.2 Introduction to Conics:
ParabolaGeneral Equation of all Conics
Latus rectum
The General equation of all Conics
Definition of a Conics
conic - a curve generated by the intersection of a plane and a circular cone
The General equation of all Conics
Definition of a Conics
conic - a curve generated by the intersection of a plane and a circular cone
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0;
Where A, B, C, D, E and F are all numbers
Parabola
The curve formed by the set of points in a plane that are all equally distant from both a given line (called the directrix) and a given point (called the focus) that is not on the line.
focus
The Vertex of the Parabola
The midpoint of a line segment between the Focus and
the Directrix
vertex 0,0
Equation of the Parabola
Depend if the parabola open to the right / left or Up and Down.
Up or Down Right / left
kyphx 42 hxpky 42
Writing the equation of the Parabola
Find the Vertex and a point on the parabola.
What Equation to Use?
Writing the equation of the Parabola
Replace h,k, x and y.
Vertex ( 1, -4)
Point ( 0, -3)
Need to solve for p. kyphx 42
Writing the equation of the Parabola
Replace h, k and p.
Vertex ( 1, -4)
Point ( 0, -3)
4
141
14)1(
4)3(4102
2
pp
p
p
Writing the equation of the Parabola
Replace h, k and p.
41
44
141
2
2
yx
yx
The Chord touching the parabola and going through the center is called Latus rectum
The Latus rectum goes through the Focus.
The Latus rectum
is 4 p
p
Find the equation of the Line tangent to the parabola at a given point
Given point (3,3): Focus (0, 2)
Equation (x - 0)2 = 0.2(y – 1)
F
Find the equation of the Line tangent to the parabola at a given point
Given point (3,3): Focus (0, 2)
Equation (x - 0)2 = 0.2(y – 1)
F 1d
2d
21 dd
Find the equation of the Line tangent to the parabola at a given point
Given point (3,3): Focus (0, 2)
Equation (x - 0)2 = 0.2(y – 1)
F 1d
2d
1019
3230
1
22
1
d
d
Find the equation of the Line tangent to the parabola at a given point
Given point (3,3): Focus (0, 2)
Equation (x - 0)2 = 0.2(y – 1)
F 1d
2d
1019
3230
1
22
1
d
d
102,0
3,3
Find the equation of the Line tangent to the parabola at a given point
Slope m =
F 1d
2d
102,0
3,3
3
101
03
1023
Find the equation of the Line tangent to the parabola at a given point
Point-slope form the line
F 1d
2d
102,0
3,33
101m
11 xxmyy
Find the equation of the Line tangent to the parabola at a given point
Point-slope form the line
3
101m
11 xxmyy
102
3
101
1011013
1011013
33
101)3(
xy
xy
xy
xy
Homework
Page 712 – 715
# 6, 12, 18, 24,
28, 34, 40, 44,
50, 56, 64, 70
Homework
Page 712 – 715
# 10, 20, 26, 42,
48, 58
