chapter 4-the ellipse · 2019-09-21 · chapter 4: the ellipse vlecture 12:introduction to ellipse...

Chapter 4: The Ellipse

SSMth1: PrecalculusScience and Technology, Engineering

and Mathematics (STEM)

Mr. Migo M. Mendoza

Chapter 4: The Ellipsev Lecture 12: Introduction to Ellipsev Lecture 13: Converting General

Form to Standard Form of Ellipse and Vice-Versa

v Lecture 14: Graphing Ellipse with Center at the Origin C (0, 0)

v Lecture 15: Graphing Ellipse with Center at C (h, k)

v Lecture 16 : The Ellipse and the Tangent Line

Lecture 12: Introduction to the Ellipse



Mr. Migo M. Mendoza

Definition of the Ellipsev An ellipse is a set of all points

P (x, y) in a plane, the sum of whose distances from two

specified fixed points F1 and F2 is a constant.

Explain the Definition of Ellipse Using the Figure Below:

Definition of the FocivThese are the two fixed

points (F1 and F2) of the ellipse.

Symbols for Foci

vF1 , F2 , to denote the foci of

the ellipse.

Definition of an Ellipsev The ellipse can also be

defined as the locus of points whose distance from

the focus is proportional to the horizontal distance from a directrix, where the ratio is

0 is less than x but x is less than 1.

Two Types of EllipsevHorizontal EllipsevVertical Ellipse

The Horizontal EllipsevWhen the foci are

on the x-axis or parallel to the x-

axis, then the ellipse is horizontal.

Something to think about…v What should be TRUEabout the coordinates of the

foci for the ellipse to be HORIZONTAL in:a) foci are on the x-axis?

b) foci are parallel to the x-axis?

Answer:a) foci are on the x-axis:

𝑥 = 𝑥|𝑥 ∈ ℝ; 𝑥 ≠ 0; 𝑖𝑓 𝑥 = 0; 𝑡ℎ𝑒𝑛 𝐹0 ≠ 𝐹1

𝑦 = 𝑦|𝑦 ∈ ℝ; 𝑦 = 0

Answer:b) foci are parallel to the x-

axis:𝑥 = 𝑥|𝑥 ∈ ℝ; 𝑥 ≠ 0; 𝑖𝑓 𝑥 = 0; 𝐹0 ≠ 𝐹1

𝑦 = 𝑦|𝑦 ∈ ℝ; 𝑦 ≠ 0

The Vertical EllipsevWhen the foci are

on the y-axis or parallel to the y-

axis, then the ellipse is vertical.

Something to think about…v What should be TRUEabout the coordinates of the

foci for the ellipse to be VERTICAL in:

a) foci are on the y-axis?b) foci are parallel to the y-axis?

Answer:a) foci are on the y-axis:

𝑥 = 𝑥|𝑥 ∈ ℝ; 𝑥 = 0

𝑦 = 𝑦|𝑦 ∈ ℝ; 𝑦 ≠ 0; 𝑖𝑓 𝑦 = 0; 𝐹0 ≠ 𝐹1

Answer:b) foci are parallel to the y-

axis:𝑥 = 𝑥|𝑥 ∈ ℝ; 𝑥 ≠ 0

𝑦 = 𝑦|𝑦 ∈ ℝ; 𝑦 ≠ 0; 𝑖𝑓 𝑦 = 0; 𝐹0 ≠ 𝐹1

The Ellipse:Parts of the Graph of the Ellipse



Mr. Migo M. Mendoza

Definition of the Axis of Symmetry

vThe line that passes through both foci is the axis of symmetry and

meets at two points called vertices.

Symbols for Vertices

vV1 ,V2 , to denote the vertices of the

ellipse.

Definition of the Major Axisv The line segment joining the

vertices and the foci is called the major axis. It is also

called as traverse axis and has a length of 2a.

The Semi-Major Axis

vThe letter a is called the semi-major axis of the ellipse.

Definition of the Minor Axisv The line segment which is a perpendicular bisector of the major axis is called minor axis. It is

also called as conjugate axis with a length of 2b.

The Semi-Minor Axisv The semi-minor axis is the

value of b in the length of minor axis of an ellipse

which is 2b.

Symbols for Covertices

vB1 ,B2 , to denote the co-vertices of the

ellipse.

Definition of the Center

vThe center of an ellipse is the intersection of the

major axis and the minor axis.

Definition of the Center

vC, to denote the center of the

ellipse.

Definition of the Directrixv It is a line such that the ratio

of distance of the points on the conic section from focus to its distance from the directrix is

constant.

Symbols for Directrices

vD1, D2, to denote the directrices of the

ellipse.

Definition of the Latera Recta� The plural form of latus

rectum, is the chord that passes through the focus, and is

perpendicular to the major axis and has both endpoints on the

curve.

Symbols for the Endpoints of Latera Recta

vE1, E2, E3, E4 to denote the endpoints of the

latera recta of the ellipse.

Other Symbols:

va, is the distance from the center to the vertex;

vb, is the distance from the center to one endpoint of the

minor axis;

Other Symbols:

vc, is the distance from the center to the focus;ve, is the value of the

eccentricity

Other Symbols:

v2a, is the length of the major axis; and

v2b, is the length of the minor axis.

Classroom Task 11:v Using the definition of

ellipse and parts of it, derive the standard

equation of horizontal ellipse with vertex at the

origin.

Standard Equation of the Horizontal Ellipse with Vertex at the Origin:

v The Standard Equation of the Horizontal Ellipse with Vertex at the Origin:

12

2

2

2

=+by

ax

Definition of the Ellipsev An ellipse is a set of all points

P (x, y) in a plane, the sum of whose distances from two

specified fixed points F1 and F2 is a constant.

The Horizontal Ellipse with Vertex at the Origin:

Representations:vLet:• P, be the point in the plane/ ellipse with

P(x, y); x = x and y = y;• F1, be one of the fixed points with F1 (c, 0);

x = c and y = 0;• F2, be one of the fixed points with F2 (-c, 0);

x = -c and y = 0;• PF1, be the distance from P(x, y)

and F1(c, 0);• PF2, be the distance from P(x, y) and

F2(-c, 0); and• k, be the sum of the distances of PF1 and

PF2 which is constant.

Something to think about…vWhat is the relationship

of the sum of the two sides of a triangle to its

third side?

The Relationship:v In a triangle, the sum of

the lengths of the two sides is GREATER

THAN the third side.

Standard Equation of the Horizontal Ellipse with Vertex at the Origin:

v The Standard Equation of the Horizontal Ellipse with Vertex at the Origin:

12

2

2

2

=+by

ax

The Horizontal Ellipse with Vertex at the Origin:

Standard Equation of the Vertical Ellipse with Vertex at the Origin:

v The Standard Equation of the Vertical Ellipse with Vertex at the Origin:

12

2

2

2

=+bx

ay

Ellipse in the Real World



Mr. Migo M. Mendoza

TED Ed Video (Nice to Know):

vLithotripsy: Ellipse in the

Medicine Field

../../TED%20Ed%20Helpful%20Videos%20in%20Teaching%20General%20Mathematics%20and%20Precalculus/What%20causes%20kidney%20stones%20-%20Arash%20Shadman.mp4

Extracorporeal Shock Wave Lithotripsy

Medicine: Medical Lithotripsy

Arts: Elliptical Whispering Gallery

Civil Engineering: Elliptical Bridge

Aeronautical Engineering: British Spitfire

Automobile: Elliptical Gears

Naval Architecture: Racing Sailboat

Astronomy: Planetary Motion

The Ellipse:Properties of the Ellipse



Mr. Migo M. Mendoza

Property Number 1 of Ellipse:

vThe length of the major axis is 2a.


vThe length of the minor axis is 2b.


vThe length of the latusrectum is

.2

ab


� The center is the intersection of the

axes.


vThe endpoints of the major axis are

called the vertices.


vThe endpoints of the minor axis are

called the co-vertices.


� The line segment joining the vertices is called the major

axis.


vThe line segment joining the co-vertices is called the minor

axis.


vThe eccentricity of the ellipse is

.10 << e

Theorem 4.2: The Eccentricity (e) of the Ellipse

v The eccentricity (e) of an ellipse is the ratio of the

undirected distance between the foci to the undirected distance between

vertices; that is:.ace =

Classroom Task 12:v We say that “circle is a special type of an ellipse.” Can you prove that circle

is a special type of an ellipse?

What do we know about their radii?

Performance Task 12:vPlease download, print

and answer the “Let’s Practice 12.” Kindly work

independently.

Lecture 12: Converting General Form of the Ellipse to Its Standard

Form and Vice-Versa



Mr. Migo M. Mendoza

What should you expect?v This section represents how

to convert general form of ellipse to its standard form

and vice-versa.

Table 4.1: Equations of EllipseCenter

Major Axis

General Form Standard Form

(0, 0) x-axis

(0, 0) y-axis

(h, k) x-axis

(h, k) y-axis

CAFCyAx <=++ ,022

CAFEyDxCyAx

<=++++ ,022

baby

ax

>=+ ,12

2

2

2

babky

ahx

>

=-

+- ,1)()(

2

2

2

2

babx

ay

>=+ 12

2

2

2CAFAyCx >=++ 022

babhx

aky

>=-

+- 1)()(

2

2

2

2

CAFEyDxAyCx >=++++ 022

Example 32:v Convert the following general

equations to standard form:

28889 22 =+ yx

Final Answerv The standard form is:

13632

22

=+yx



225,14925 22 =+ yx


12549

22

=+yx



052162443 22 =+-++ yxyx


13)2(

4)4( 22

=-

++ yx

Conclusion 1 about the Ellipse:

vWhen the radius of the ellipse is of positive sign, then the ellipse

exists.


equations to standard form:0110245469 22 =+-++ yxyx


5)2(6)3(9 22 -=-++ yx

Something to think about…

vWhat can you observe on the right side of the

equation? What can you conclude?


vWhen the radius of the ellipse is of negative sign, then the ellipse

does not exist.


equations to standard form:0180247249 22 =++++ yxyx


0)3(4)4(9 22 =+++ yx

Something to think about…

vWhat can you observe on the right side of the

equation? What can you conclude?


vWhen the radius of the ellipse is of zero value,

then the ellipse will degenerate to a point.

Example 37:v Convert the following standard

form to general form:

19)3(

25)2( 22

=-

+- xy

Final Answerv The general form is:

03636150925 22 =+--+ yxyx

Example 38:v Convert the following standard

form to general form:

136)1(

100)1( 22

=+

+- yx

Final Answerv The general form is:

0464,32007210036 22 =-+-+ yxyx



independently.

Lecture 13: Graphing an Ellipse with Center at the Origin



Mr. Migo M. Mendoza

Learning Expectation:v This section presents how

to graph an ellipse and how to determine the parts of an ellipse where the center is at

the origin.

Table 4.2 : Parts of the Graph of Ellipse with Center at the Origin

Standard Equation

Foci VerticesCo-

verticesEndpoints of Latera Recta

Directrix

F1 (c, 0)

F2 (-c, 0)

V1 (a, 0)

V2 (-a, 0)

B1 (0, b)

B2 (0, -b)

F1 (0, c)

F2 (0, -c)

V1 (0, a)

V2 (0, -a)

B1 (b, 0)

B2 (-b, 0)

÷÷ø

öççè

æ--÷÷

ø

öççè

æ-

÷÷ø

öççè

æ-÷÷

ø

öççè

æ

abcE

abcE

abcE

abcE

2

4

2

3

2

2

2

1

,,

,,

cax2

±=

÷÷ø

öççè

æ--÷÷

ø

öççè

æ-

÷÷ø

öççè

æ-÷÷

ø

öççè

æ

cabEc

abE

cabEc

abE

,,

,,

2

4

2

3

2

2

2

1

cay2

±=

12

2

2

2

=+by

ax

12

2

2

2

=+bx

ay

Example 39:v Sketch and discuss the following

equation of an ellipse:

225925 22 =+ yx

Example 40:v Find the equation of the ellipse

with center at C (0, 0), length of major axis is 10 units, and a

focus at F1 (4, 0). Identify the parts of the ellipse and sketch its

graph.

Final Answer:v The equation of the ellipse is:

.1925

22

=+yx

Example 41:v Find the equation of the ellipse

with center at C (0, 0), vertices at

and eccentricity

Identify the parts of the ellipse and sketch its graph.

),0,4(± .23

=e


.1416

22

=+yx



independently.

Lecture 14: Graphing an Ellipse with Center at C (h, k)



Mr. Migo M. Mendoza

Learning Expectation:vThis section presents

how to graph an ellipse and how to determine the parts of an ellipse.

Table 4.2 : Parts of the Graph of Ellipse with Center at the Origin

Standard Equation

Foci VerticesCo-

verticesEndpoints of Latera Recta

Directrix

𝑥 − ℎ 1

𝑎1 +𝑦 − 𝑘 1

𝑏1 = 1𝐹0 ℎ + 𝑐, 𝑘𝐹1 (ℎ − 𝑐, 𝑘)

𝑉0 ℎ + 𝑎, 𝑘𝑉1(ℎ − 𝑎, 𝑘)

𝐵0 ℎ, 𝑘 + 𝑏𝐵1(ℎ, 𝑘 − 𝑏)

𝐸0 ℎ + 𝑐, 𝑘 +𝑏1

𝑎

𝐸1 ℎ − 𝑐, 𝑘 +𝑏1

𝑎

𝐸@ ℎ + 𝑐, 𝑘 −𝑏1

𝑎

𝐸A ℎ − 𝑐, 𝑘 −𝑏1

𝑎

𝑥 = ℎ ±𝑎1

𝑐

𝑦− 𝑘 1

𝑎1 +𝑥 − ℎ 1

𝑏1 = 1𝐹0 ℎ , 𝑘 + 𝑐𝐹1 (ℎ, 𝑘 − 𝑐)

𝑉0 ℎ, 𝑘 + 𝑎𝑉1(ℎ, 𝑘 − 𝑎)

𝐵0 ℎ + 𝑏, 𝑘𝐵1(ℎ − 𝑏, 𝑘)

𝐸0 ℎ +𝑏1

𝑎, 𝑘 + 𝑐

𝐸1 ℎ −𝑏1

𝑎, 𝑘 + 𝑐

𝐸@ ℎ +𝑏1

𝑎 , 𝑘 − 𝑐

𝐸A ℎ −𝑏1

𝑎, 𝑘 − 𝑐

𝑦 = 𝑘 ±𝑎1

𝑐

Example 42:v Sketch and discuss the following

equation of an ellipse:

0464,32007210036 22 =-+-+ yxyx

Example 43:v Find the equation of the ellipse with center at C (-4, 7),

a focus at F1 (-4, 11) and a vertex at V2 (-4, 12). Identify

the parts of the ellipse and sketch its graph.


19)4(

25)7( 22

=+

+- xy

.0616126200925 22 =+-++ yxyxor

Example 44:v Find the equation of the ellipse with

center at C (2, -3), vertices at V1 (7, -3) and V2 (-3, -3), and eccentricity of e = 3/5. Identify the

parts of the ellipse and sketch its graph.


116)3(

25)2( 22

=+

+- yx

.0111150642516 22 =-+-+ yxyxor



independently.

Lecture 15: Tangent to the Ellipse



Mr. Migo M. Mendoza

Tangential to the Ellipse

vA tangent to the ellipse is a line that

touches the ellipse at just one point.

Tangent to the Ellipse

The Equation of the LinevThe equation of the line can be

determined using the formula:

222 bmamxy +±=

Example 45:v Find the equation of the

tangent to the ellipse and the line

passes at a point PT (12, 3). Sketch its graph.

364 22 =+ yx

Final Answer:vThe equation of the tangent

line is:

.01532 =-- yx

Example 46:v Find the point on the ellipse

which is the closest, and which is the farthest point from the line

Sketch the graph.

365 22 =+ yx

.03052 =+- yx

Final Answer:

v The closest point is (-4, 2) and the farthest point is

(4, -2) to the line .01532 =-- yx



independently.

chapter 4-the ellipse · 2019-09-21 · chapter 4: the ellipse vlecture 12:introduction to ellipse...

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