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conic section and its types and general parts (1) Parabola, parts and its properties
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Conic Sections Parabola (Part 1)
Irish Anne Ubalde
January 29, 2015
APOLLONIUS OF PARGA
What is a Conic Section?
Conic Section
Sections formed when planes cut a right circular cone of two nappes.
Conic Sections
General Parts of Conics
1. Focus is the fixed point F
2. Directrix
is the fixed line D
3. Eccentricity is the positive constant ratio (ratio from the segment connecting F to the conic section and from D to cs)
If e=1, parabola
e1, hyperbola
Parabola Definition:
Is the locus of a point that moves in a plane so that its distance from a fixed point is equal to its distance from a fixed line. Its eccentricity is 1.
Geogebra
Parts of Parabola
A. Focus
B. Directrix
C. Vertex
D. Axis of symmetry
E. Focal Chord
F. Latus Rectum
Geogebra
Important Measures of Parabola
1. a = distance from F to V
= distance from V to D
2. 2a = distance from F to D
= distance from F to an end of LR
3. 4a = distance from one end of LR to the other
4. = eccentricity
Geogebra
Standard Forms of Parabola
When V (0,0)
1. Axis of Symmetry on y-axis a. Upward opening b. Downward opening = =
Geogebra
2. Axis of Symmetry on x-axis
a. Right opening b. Left opening = =
Example 1: Determine the opening of each parabola
1. 2 = 4
2. 2 = 4
3
3. 2 = 6y
4. 2 = 8
right
downward
upward
left
5. 2 16 = 0 right
6. 22 + 30 = 0 downward
Example 3: Reduce 2 + 12 = 0 to standard form and
determine the following:
a. Opening of the parabola
b. Vertex
c. Focus
d. Equation of directrix
e. Ends of Latus Rectum
downward
(0,0)
= =
F(0,-3)
=
, &(, )
Example 2:
Sketch the graph of the parabola 2 16 = 0
a. 2 16 = 0
Solution: 2 = 16 Transform in s.f 2 = 4
Opening is right Vertex at (0,0)
4 = 16 length of latus rectum
2 = 8 length of (a) F to one end of LR; (b) F to D
= 4 length of (a) F to V; (b) V to D
Answer: Opening is right V: (0,0) F: (4,0) LR: (4,8) , (4, 8) D: = 4
Seatwork:
P. 100 # 2
Sketch the graph of the parabola 2 = 20.
Determine the opening, locate the vertex, focus, ends of latus rectum and the equation of directrix
Standard Forms of Parabola When V (h,k)
3. Axis of Symmetry vertical
a. Upward opening b. Downward opening ( )2= 4 ( )2= 4( )
Geogebra
4. Axis of Symmetry horizontal a. Right opening b. Left opening ( )2= 4 ( )2= 4( )
Proof:
(, )
(, )
=
Example 3:
Reduce 2 + 16 32 = 0 to s.f, find the direction of opening, vertex, focus, endpoints of latus rectum, determine the equation of the directrix and draw the parabola.
Exercise:
Reduce 2 4 + 8 20 = 0 to s.f, find the direction of opening, vertex, focus, endpoints of latus rectum, determine the equation of the directrix and draw the parabola.