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Conic Sections

Math Response

What is 23x2

Type answer here

Correct Answer: 456

Deg^°

Rad

Objectives:

At the end of the lesson, the students should be able to:

✓ define an ellipse
✓ name the parts and properties of an ellipse
✓ write the equation of an ellipse
✓ Illustrate the different types of conic sections: parabola,

ellipse, circle, hyperbola, and degenerate cases

Ellipse

Ellipses are formed when the plane
intersects the one cone at an angle

other than 90°.

An ellipse is a set of points in

a plane whose sum of

distances from two fixed

points 𝐹1 and 𝐹2 is constant.

What is an ellipse?

What are the properties of ellipse?

Each of the two fixed points 𝐹1 and 𝐹2 is a

focus of the ellipse.

Multiple Choice

Which of the following is true?

• wd

• d

• d

• d

What are the properties of ellipse?

The line passing through the foci (plural of focus) of

an ellipse is called the principal axis.

What are the properties of ellipse?

The two points on the ellipse that lie on the

principal axis are the vertices.

What are the properties of ellipse?

The midpoint of the two vertices is the center

of the ellipse.

What are the properties of ellipse?

The line segment joining the vertices is called the

major axis.

What are the properties of ellipse?

The line segment that passes through the center and is

perpendicular to the major axis whose endpoints are on the ellipse

is called the minor axis.

Take note of the following notations:

• a is the distance from the

center to a vertex

• b is the distance from the

center to a co-vertex, and

• c is the focal distance

Standard Equation of an Ellipse with center at (0,0)

Standard Equation of an Ellipse with center at (h,k)

Example 1:

Find the properties of an ellipse whose equation is defined by

𝑥2

16+

𝑦2

4= 1.

To identify which of these is our 𝑎 and 𝑏, remember that in ellipse, 𝑎 > 𝑏.
So, in this equation, the values of
𝑎2= 16

⇒ 𝑎 =

16

⇒ a = 4

𝑏2= 4

⇒ 𝑏 =

4

⇒ b = 2

and

𝑥2

16+ 𝑦2

4= 1

Standard Equation

𝑥2

42 + 𝑦2

22 = 1

where:

𝑎 = 4

𝑏 = 2

To solve for c :

𝑏2+ 𝑐2= 𝑎2

𝑐2= 𝑎2− 𝑏2

𝑐2= 42− 22

𝑐2= 12
𝑐 =

12

Example 1:

Find the properties of an ellipse whose equation is defined by

𝑥2

16+

𝑦2

4= 1.

Properties

ሻ

Center: 𝐶(0, 0

Vertices: 𝑉1 −4, 0 ; 𝑉2(4, 0ሻ
Co-vertices: C𝑉1 0, 2 ;𝐶𝑉2(0, −2ሻ

Length of the major axis:

𝟐𝒂 ⇒ 2 4 = 8 𝑢𝑛𝑖𝑡𝑠

Principal axis: 𝒙 − 𝒂𝒙𝒊𝒔

Foci: 𝐹1 −2 3, 0 ; 𝐹2(2 3, 0)

Length of the minor axis:

𝟐𝒃 ⇒ 2 2 = 4 𝑢𝑛𝑖𝑡𝑠

Example 2:

Find the properties of an ellipse whose equation is defined by

(𝑥−1ሻ2

49
+

(𝑦+3ሻ2

64
= 1.

To identify which of these is our 𝑎 and 𝑏, remember that in ellipse, 𝑎 > 𝑏.
So, in this equation, the values of
𝑎2= 64

⇒ 𝑎 =

64

⇒ a = 8

𝑏2= 49

⇒ 𝑏 =

49

⇒ b = 7

and

𝑥2

49+ 𝑦2

64= 1

Standard Equation

𝑥2

72 + 𝑦2

82 = 1

where:

𝑎 = 8

𝑏 = 7

To solve for c :

𝑏2+ 𝑐2= 𝑎2

𝑐2= 𝑎2− 𝑏2

𝑐2= 82− 72

𝑐2= 15
𝑐 =

15

Example 2:

Find the properties of an ellipse whose equation is defined by

(𝑥−1ሻ2

49
+

(𝑦+3ሻ2

64
= 1.

Properties

ሻ

Center: 𝐶(1, −3

Vertices: 𝑉1 1, 5 ; 𝑉2(1, −11ሻ
Co-vertices: C𝑉1 −6, −3 ;𝐶𝑉2(8, −3ሻ

Length of the major axis:

𝟐𝒂 ⇒ 2 8 = 16 𝑢𝑛𝑖𝑡𝑠

Principal axis: y−𝒂𝒙𝒊𝒔

Foci: 𝐹1 1, −3 +

15 ; 𝐹2(1, −3 −

15)

Length of the minor axis:

𝟐𝒃 ⇒ 2 7 = 14 𝑢𝑛𝑖𝑡𝑠

Example 3:

Find the properties of an ellipse whose equation is defined by

(𝑥+3ሻ2

36
+

(𝑦−2ሻ2

9
= 1.

Example 3:

Find the properties of an ellipse whose equation is defined by

(𝑥+3ሻ2

36
+

(𝑦−2ሻ2

9
= 1.

Example 4:

Find the equation of the ellipse whose foci is at 𝐹1 0,3 and 𝐹1 0, −3 , and has a

minor axis of length 8.

Example 4:

Find the equation of the ellipse whose foci is at 𝐹1 0,3 and 𝐹1 0, −3 , and has a minor

axis of length 8.

Any more questions?