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Chapter 1: Introduction to Conic Section SSMth1: Precalculus Science and Technology, Engineering and Mathematics (STEM) Mr. Migo M. Mendoza

Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

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Page 1: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Chapter 1: Introduction to Conic Section

SSMth1: Precalculus

Science and Technology, Engineering

and Mathematics (STEM)

Mr. Migo M. Mendoza

Page 2: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Tell me, what do you see?

Page 3: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

A Double-Napped Circular Cone

Page 4: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

A Double-Napped Circular Cone

Page 5: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Conic Sections

Page 6: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

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.

Page 7: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Parts of a Double-Napped Circular Cone

Central Axis Generators Vertex

Upper and Lower Nappes

Vertex Angles Circular Base

Page 8: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

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.

Page 9: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

The Generators

These are the diagonal sides of the double-napped cone.

Also, it is the line that rotates about the fixed point.

Page 10: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

The Vertex

It is the point at the center of a double-napped

cone. Also, it is a fixed point.

Page 11: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

The Upper and Lower Nappes

These are the lateral surfaces of the double-right

circular cone.

Page 12: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

The Vertex Angle

It is the angle between the central axis and the

generator. It is denoted by α.

Page 13: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Something to think about…

What will happen if a plane intersects a

double-napped circular cone?

Page 14: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Figure 1: Circle

Page 15: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

When Does a Circle Formed?

A circle is produced when the plane passes

through one nappe only, perpendicular to the

central axis.

Page 16: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

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.

Page 17: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Figure 2: Ellipse

Page 18: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

When Does an Ellipse Formed?

An ellipse is produced when the plane passes through one

nappe only, between the generator and perpendicular.

Page 19: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

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.

Page 20: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Figure 3: Parabola

Page 21: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

When Does a Parabola Formed?

A parabola is produced when the plane passes

through one nappe parallel to the generator.

Page 22: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

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.

Page 23: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Figure 4: Hyperbola

Page 24: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

When Does a Hyperbola Formed?

A hyperbola is produced when the plane passes through

both nappes, between the central axis and the generator.

Page 25: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

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.

Page 26: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

The Conic Sections

The CircleThe EllipseThe ParabolaThe Hyperbola

Page 27: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Something to think about…

What have you observed on how four

conic sections were formed?

Page 28: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

What have you Observed?

Page 29: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Take Note:

The basic four conic sections can only be produced when the plane does NOT pass

through the vertex.

Page 30: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Something to think about…

What will happen if the plane passes

through the vertex?

Page 31: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Figure 5: Degenerated Circle

Page 32: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Case 1: Degenerated Circle

A circle will degenerate into a

point.

Page 33: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Figure 6: Degenerated Ellipse

Page 34: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Case 2: Degenerated Ellipse

An ellipse will degenerate into a

point.

Page 35: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Figure 7: Degenerated Parabola

Page 36: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Case 3: Degenerated Parabola

A parabola will degenerate into a

single line.

Page 37: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Figure 8: Degenerated Hyperbola

Page 38: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Case 4: Degenerated Hyperbola

A hyperbola will degenerate into two intersecting lines.

Page 39: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

The Three Degenerate Conic Sections

1. A Point2. A Single Line

3. Two Intersecting Lines

Page 40: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

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?

Page 41: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

The Non-Degenerate Conic Sections

1. Circle2. Ellipse

3. Parabola4. Hyperbola

Page 42: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Summary of the Four Basic Conic Sections

How Conic Sections were

Formed

Page 43: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Something to think about…

Is it possible to determine the type of conic sections we have if the only given is its

equation?

Page 44: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Classroom Task 1:

Determine the type of conic section that each general equation will

produce:

Page 45: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Classroom Task 1:

04324264.

04424842.

03624844.

01118699.

22

22

22

22

yxyxyxd

yxyxc

yxyxyxb

yxyxa

Page 46: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Take Note:

022 FEyDxCyBxyAx

The graph of the second-degree equation of the form

is determined by the values of

.42 ACB

Page 47: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Something to think about…

Why do you think our four basic conic sections have the graph of the second-degree

equation?

Page 48: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Something to think about…

What do you still remember about

?ACB 42

Page 49: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

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.

Page 50: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

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

Page 51: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Something to think about…

What is eccentricity?

Page 52: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

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.

Page 53: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Understanding Eccentricity

Page 54: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Example 1:

Determine the type of conic section that each general

equation will produce:

01118699. 22 yxyxa

Page 55: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Final Answer:

Take note that in

B = 0 and A = C. Thus, the conic section is a circle.

01118699 22 yxyx

Page 56: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Example 2:

Determine the type of conic section that each general

equation will produce:

03624844. 22 yxyxyxb

Page 57: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Final Answer:

Thus, the conic section for

is a parabola.03624844 22 yxyxyx

Page 58: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Example 3:

Determine the type of conic section that each general

equation will produce:

04424842. 22 yxyxc

Page 59: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Final Answer:

Take note that in

B = 0 and A ≠ C. Thus, the conic section is an ellipse.

04424842 22 yxyx

Page 60: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Example 4:

Determine the type of conic section that each general

equation will produce:

04324264. 22 yxyxyxd

Page 61: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Final Answer:

Thus, the conic section for

is a hyperbola

04324264 22 yxyxyx

Page 62: Chapter 1: Introduction to Conic Section · PDF fileThe Ellipse The Parabola The Hyperbola. 6RPHWKLQJWRWKLQNDERXW

Performance Task 1:

Please download, print

and answer the “Let’s

Practice 1.” Kindly work

independently.