Post on 05-Feb-2018
transcript
Chapter 1: Introduction to Conic Section
SSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
Tell me, what do you see?
A Double-Napped Circular Cone
A Double-Napped Circular Cone
Conic Sections
A Double-Napped Circular Cone
It is the shape formed when two congruent cones put on top of each other, their tips touching and their
axes aligned, with each are extending indefinitely away from their tips.
Parts of a Double-Napped Circular Cone
Central Axis Generators Vertex
Upper and Lower Nappes
Vertex Angles Circular Base
The Central Axis
It is the vertical line down the middle of a double-napped
cone. Also, it is the line that remain at fixed.
The Generators
These are the diagonal sides of the double-napped cone.
Also, it is the line that rotates about the fixed point.
The Vertex
It is the point at the center of a double-napped
cone. Also, it is a fixed point.
The Upper and Lower Nappes
These are the lateral surfaces of the double-right
circular cone.
The Vertex Angle
It is the angle between the central axis and the
generator. It is denoted by α.
Something to think about…
What will happen if a plane intersects a
double-napped circular cone?
Figure 1: Circle
When Does a Circle Formed?
A circle is produced when the plane passes
through one nappe only, perpendicular to the
central axis.
Relationship of Angle α and Angle β
When the angle made by the plane and the central axis (β)
is exactly 90°, the conic section is a circle.
Figure 2: Ellipse
When Does an Ellipse Formed?
An ellipse is produced when the plane passes through one
nappe only, between the generator and perpendicular.
Relationship of Angle α and Angle β
When the angle made by the plane and the central axis (β)
is greater than the vertex angle (α) the conic section is
an ellipse.
Figure 3: Parabola
When Does a Parabola Formed?
A parabola is produced when the plane passes
through one nappe parallel to the generator.
Relationship of Angle α and Angle β
When the angle made by the plane and the central axis (β) is equal to the vertex angle (α) the
conic section is a parabola.
Figure 4: Hyperbola
When Does a Hyperbola Formed?
A hyperbola is produced when the plane passes through
both nappes, between the central axis and the generator.
Relationship of Angle α and Angle β
When the angle made by the plane and the central axis (β) is
less than the vertex angle (α) the conic section is a hyperbola.
The Conic Sections
The CircleThe EllipseThe ParabolaThe Hyperbola
Something to think about…
What have you observed on how four
conic sections were formed?
What have you Observed?
Take Note:
The basic four conic sections can only be produced when the plane does NOT pass
through the vertex.
Something to think about…
What will happen if the plane passes
through the vertex?
Figure 5: Degenerated Circle
Case 1: Degenerated Circle
A circle will degenerate into a
point.
Figure 6: Degenerated Ellipse
Case 2: Degenerated Ellipse
An ellipse will degenerate into a
point.
Figure 7: Degenerated Parabola
Case 3: Degenerated Parabola
A parabola will degenerate into a
single line.
Figure 8: Degenerated Hyperbola
Case 4: Degenerated Hyperbola
A hyperbola will degenerate into two intersecting lines.
The Three Degenerate Conic Sections
1. A Point2. A Single Line
3. Two Intersecting Lines
Something to think about…
If two intersecting lines, a single line, and a point constitute
the degenerate conic sections, then what are the non-
degenerate conic sections?
The Non-Degenerate Conic Sections
1. Circle2. Ellipse
3. Parabola4. Hyperbola
Summary of the Four Basic Conic Sections
How Conic Sections were
Formed
Something to think about…
Is it possible to determine the type of conic sections we have if the only given is its
equation?
Classroom Task 1:
Determine the type of conic section that each general equation will
produce:
Classroom Task 1:
04324264.
04424842.
03624844.
01118699.
22
22
22
22
yxyxyxd
yxyxc
yxyxyxb
yxyxa
Take Note:
022 FEyDxCyBxyAx
The graph of the second-degree equation of the form
is determined by the values of
.42 ACB
Something to think about…
Why do you think our four basic conic sections have the graph of the second-degree
equation?
Something to think about…
What do you still remember about
?ACB 42
DiscriminantIn a quadratic equation, the
discriminant helps tell you the number of real solutions to a quadratic equation.
The expression used to find the discriminant is the expression located
under the radical in the quadratic formula.
Table 1: Graphs of Second-Degree Equation
Conic SectionValue of the Discriminant
Eccentricity
Circle B = 0 or A = C
Parabola
Ellipse B = 0 or A ≠ C
Hyperbola
;042 ACB
042 ACB
042 ACB
042 ACB
0e
1e
10 e
1e
Something to think about…
What is eccentricity?
Eccentricity
The eccentricity, denoted by e or ε,
is a parameter associated with every conic section. It can be thought of as a
measure of how much the conic section deviates from being circular.
Understanding Eccentricity
Example 1:
Determine the type of conic section that each general
equation will produce:
01118699. 22 yxyxa
Final Answer:
Take note that in
B = 0 and A = C. Thus, the conic section is a circle.
01118699 22 yxyx
Example 2:
Determine the type of conic section that each general
equation will produce:
03624844. 22 yxyxyxb
Final Answer:
Thus, the conic section for
is a parabola.03624844 22 yxyxyx
Example 3:
Determine the type of conic section that each general
equation will produce:
04424842. 22 yxyxc
Final Answer:
Take note that in
B = 0 and A ≠ C. Thus, the conic section is an ellipse.
04424842 22 yxyx
Example 4:
Determine the type of conic section that each general
equation will produce:
04324264. 22 yxyxyxd
Final Answer:
Thus, the conic section for
is a hyperbola
04324264 22 yxyxyx
Performance Task 1:
Please download, print
and answer the “Let’s
Practice 1.” Kindly work
independently.