Math 10th grade
LEARNING UNIT
Let us discover new ways
and use the Cartesian
Plane.
S/K
LEARNING OBJECT
Description of the Ellipse
SKILL 1: Analyze experiments where the reflective
property of the Ellipse is displayed
SKILL 2: Based on observations, describe the
reflective property of the Ellipse
SKILL 3: Identify the application of the reflective
property of ellipses when operating with lithotripters
SKILL 4: Identify the application of the reflective
property of ellipses in architecture
SKILL 5: Identify the application of the reflective
property of ellipses in communication
SKILL 6: Identify the application of the reflective
property of ellipses in astronomy
SKILL 7: Identify the application of the reflective
property when using elliptical mirrors
SKILL 8: Identify the use of elliptical shapes in artistic
creations
SKILL 9: Identify the process for the construction of
the ellipse by using a ruler and compass
SKILL 10: Create ellipses using a ruler and compass
SKILL 11: Interpret each of the steps used in the
construction of ellipses using a ruler and compass
SKILL 12: Identify the basic elements of ellipses such
as foci, focal axis, vertices, major axis, center, minor
axis, linear eccentricity, etc.
SKILL 13: Identify the distance of any point of the
ellipse to the foci
SKILL 14: Identify the sum of the distances of any
points to the foci
SKILL 15: Characterize each of the points that make
up the Ellipse
SKILL 16: Define “Ellipse”
SKILL 17: Create ellipses from their basic elements
SKILL 18: Represent ellipses in the Cartesian Plane
with foci on the horizontal axis that are equidistant
from the point of origin
SKILL 19: Recognize the equation creation process of
an the ellipse with foci on horizontal axis that are
equidistant from the point of origin
Language
Socio cultural context of
the LO
Curricular axis
Standard competencies
Background Knowledge
Basic Learning
Rights
English Review topic
Vocabulary box
SKILL 20: Using an equation, represent ellipses with
foci on the horizontal axis that are equidistant from
the point of origin
SKILL 21: Represent ellipses with foci located at any
point on the plane
SKILL 22: Recognize the process of creating ellipses
with foci located at any point on the plane
SKILL 23: Using an equation, represent ellipses with
foci located at any point on the plane
SKILL 24: Characterize an ellipse based on its
equation
SKILL 25: Build an ellipse on the Cartesian plane from
its equation or from its basic elements
SKILL 26: Find the equation of an ellipse by
identifying its representation on the Cartesian Plane
SKILL 27: Find the equation of an ellipse based on the
description of some of its elements
English
Classroom, student’s family context.
Spatial thinking and geometric systems
Identify the characteristics and properties of the
conical figures (ellipses, parabolas, hyperbolas) and
use their properties in problem solving.
Graphics in the Cartesian plane, basic algebraic
operations.
Recognize geometric properties that define different
types of conic elements (parabolae, ellipses, and
hyperbolae) on the plane and use them to find the
general equations for these types of curves.
The author's "we", passive voice
NAME: _________________________________________________
GRADE: ________________________________________________
Introduction
Guillermo visits a scientific laboratory his uncle invited him to, enters a
room that has the shape of an egg, and observes a scientist who is
playing with a flashlight. He chooses a point on the room floor and turns
on the flashlight projecting beams of light and changing the angle
between the flashlight and the floor.
The funny thing is that despite the angle change, when the beam of
light bounces, it always ends up in the same place of the room, another
marked spot.
What property does this room have for this to happen?
Objectives
1. Describe the reflective property of the ellipse and identify its use
2. Using a ruler and compass, represent an ellipse by recognizing
geometric construction strategies
3. Conceptualize an ellipse by identifying its characteristics as a locus
4. Use equations to represent ellipses on the Cartesian Plane
Activity 1
SKILL 1: Analyze experiments where the reflective property of the
Ellipse is displayed
SKILL 2: Based on observations, describe the reflective property of the
Ellipse
SKILL 3: Identify the application of the reflective property of ellipses
when operating with lithotripters
SKILL 4: Identify the application of the reflective property of ellipses in
architecture
SKILL 5: Identify the application of the reflective property of ellipses in
communication
SKILL 6: Identify the application of the reflective property of ellipses in
astronomy
SKILL 7: Identify the application of the reflective property when using
elliptical mirrors
SKILL 8: Identify the use of elliptical shapes in artistic creations
Reflective property of the ellipse
If from one of the foci a beam of light is emitted, which is reflected in
the interior of the ellipse, the reflected ray will pass through the other
foci, such as seen on Image No. 1
1
Image No. 1
Foci
Applications of the reflective property of the ellipse
Lithotripter
A lithotripter is the instrument used in medicine to disintegrate kidney
stones in patients. The lithotripter consists of a half ellipsoid instrument
(an ellipse viewed in 3D. Think of it as an egg) filled with water and an
intra-aquatic wave generator located in one of the foci of the ellipsoidal
reflector. (Garzòn, 2013)
This is to say that the reflective property of the ellipse ensures that the
waves emitted from the foci, located on the outside of the body of the
patient, then arrive at another foci where the kidney stones to be
disintegrated are located. Look at image No. 2.
1
Image
retrieved
from:
http://www.monografias.com/trabajos82/definicion-grandesconicas/image026.jpg, researched on 03-27-2016
2
Architecture
Image No. 2
The reflective property of the ellipse is used in architecture to conduct
sound. For example, an enclosure with an ellipsoid shape ensures that a
sound emitted from one of the foci will only be heard in the place where
the second foci is located, while nothing is heard in the remaining space.
Examples of these constructions are: the Torco building (Image No. 3),
the Capilla de los Secretos that can be found in the Desierto de los
Leones in Mexico DF, Saint Paul’s cathedral in London, and the National
Statutory Hall of the United States Capitol.
Astronomy
One of the applications of the ellipse and the reflective property is the
discovery of the elliptical orbit, in which the planets portray an elliptical
orbit, where one of the foci is the sun (discovery made by the German
astronomer Johannes Kepler). Furthermore, the reflective property helps
in designing precision microscopes that allow to play with converging
lenses and achieve greater precision.
Did you know that...?
At the time of the astronomer Nicolaus Copernicus it was believed that the orbits
in which the planets revolved around the Sun were circular.
Communication
This property is used to allow communication between two specific
points. This is how some satellites are located in ellipses’ foci in such a
way that communication can be established between them and not
2
Image retrieved from: http://www.bdigital.unal.edu.co/39621/1/angelicalorenagarzon.2013.pdf
(Garzòn, 2013)
dispersed into other elements of space, just like what was mentioned for
architecture.
3
Elliptical Mirrors
Image No. 3
Elliptical mirrors that take the form of an ellipse have the function of
emitting light from one of the foci, to transmit it to the other. Elliptical
mirrors subject to the reflective property are used in geometrical optics,
focal systems, ellipsoidal image projectors, construction of telescopes,
and in some car headlights.
Learning activity:
1. Look for elements in your surroundings (school, neighborhood,
and home) that have an elliptical shape. Take a picture and with
your teachers help, do a picture presentation.
2. Identify the process for the construction of an ellipse by using a
ruler and compass
3. Create ellipses using a ruler and compass
4. Analyze each of the steps used in the construction of ellipses using
a ruler and compass
Activity 2.
Creating ellipses using a ruler and compass
1. Trace the segments representing the major and minor axes.
(Image No. 4a)
2. Draw a circle with a diameter equal to the major axis and the
other with a diameter equal to the minor axis. (Image No. 4b)
3
Image retrieved from: http://www.bdigital.unal.edu.co/39621/1/angelicalorenagarzon.2013.pdf
(Garzòn, 2013)
Image No. 4a Image No. 5b
3. Through the center of the ellipse, trace radii that cut the two
circles at points P and Q. (Image No. 5a)
4. At points P, parallel lines to the minor axis are drawn. At points Q
parallel lines to the major axis are drawn. The points where the
parallel lines of the two previous steps meet will be points in the
ellipse. (Image 5b)
Image No. 5a Image No. 5b
The more radii are drawn, the more ellipse points there will be. (Image
No. 6)
Image No. 6
Learning activity
1. Using aruler and compass, build the following ellipses:
a) Larger radius equal to 8 cm and minor radius equal to 10 cm.
b) Larger radius equal to 14 cm and minor radius equal to 12 cm.
Activity 3.
SKILL 12: Identify the basic elements of ellipses such as foci, focal axis,
vertices, major axis, center, minor axis, linear eccentricity, etc.
SKILL 13: Identify the distance of any point of the ellipse to the foci
SKILL 14: Identify the sum of the distances of any points to the foci
SKILL 15: Characterize each of the points that make up the Ellipse
SKILL 16: Define “Ellipse”
SKILL 17: Create ellipses from their basic elements
Identifying an ellipse
Carefully observe the following image where the elements that make up
an ellipse are shown.
Image No. 7
Vertex, Major axis, Foci, Center, Minor axis, Foci, vertex
4
Focal Axis: The line that passes through the foci of the ellipse.
Vertices: The points where the focal axis intersects the ellipsis.
Major Axis: The portion of the focal axis between the vertices.
Center: Midpoint of the segment joining the foci
Normal Axis: The line that passes through the center and is
perpendicular to the focal axis.
● Minor Axis: The segment on the normal axis that intersects the
ellipse on two points.
●
●
●
●
●
4
Image retrieved from: http://quiz.uprm.edu/pc_cb/elipse/1.png
● Linear Eccentricity: The distance that separates the foci of the
ellipse; hereinafter the foci will be referred to as F1 and F2,
therefore the linear eccentricity will be equal to 2F.
Distance between foci and points on an ellipse
In the following image, there is an ellipse where the distances are found
from the foci to a point. Note carefully:
5
Image No. 8
Distance from F1 to p1 is: 1.58
Distance from F2 to p1 is: 0.92
As you can see they are different, but the sum will be:
F1p1+F2p1= 2.5
Now, we repeat this procedure for point p2
Image No. 9
Distance from F1 to p2 is: 0.54
Distance from F2 to p2 is: 1.96
As you can see they are different, but the sum will be: F1p2+F2p2= 2.5
Note that the sum of the distances found for p1 and p2 are the same.
This is a property that only the ellipse has and on which its definition is
built:
Definition of Ellipse
An ellipse is the locus of points that move on a plane, such that the
sum of their distances to two fixed points (foci) of that plane is always
equal to a constant.
5
Images 5 and 6 designed by the author
Example:
1. Find the ellipse whose minor axis measures 4 and
measures 6.
Major axis: 6
Minor axis: 4
major axis
Major axis: 6
Minor axis: 4
As you can see, for this situation there are two correct solutions. The
ellipses are in essence the same, their difference lies in if the major axis
is horizontal or vertical.
Learning activity:
1. Complete using the missing words:
a) The
_________
axis
is
the
on
the
_____ axis that intersects the ellipse in _____ points.
_
b) The ____________ axis is the _________ that passes through
the center and is _________ to the focal axis.
2.
Answer true or false:
a) An ellipse has a minor axis and two major axes ___________
b) If the major and the minor axes of the ellipse are the same,
then it would be a circumference. ________
Activity 4.
SKILL 18: Represent ellipses in the Cartesian Plane with foci on the
horizontal axis that are equidistant from the point of origin
SKILL 19: Recognize the equation creation process of an the ellipse with
foci on horizontal axis that are equidistant from the point of origin
SKILL 20: Using an equation, represent ellipses with foci on the
horizontal axis that are equidistant from the point of origin
SKILL 21: Represent ellipses with foci located at any point on the plane
SKILL 22: Recognize the process of creating ellipses with foci located at
any point on the plane
SKILL 23: Using an equation, represent ellipses with foci located at any
point on the plane
SKILL 24: Characterize an ellipse based on its equation
SKILL 25: Build an ellipse on the Cartesian plane from its equation or
from its basic elements
SKILL 26: Find the equation of an ellipse by identifying its
representation on the Cartesian Plane
SKILL 27: Find the equation of an ellipse based on the description of
some of its elements
Ellipse with foci on the horizontal axis
Next, we will create an ellipse’s equation whose foci are on the x axis
and equidistant from the origin point (0,0).
Image No. 10
To create the equation the definition of ellipse will be used, as we know
that the sum of distances from a point to the foci will be a constant.
Take into account that for the demonstration, the point to be taken will
be on the y axis, therefore:
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐹1𝑝 = 𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝐹2𝑝 = 𝑎
√(𝑥 − 𝑐)2 + 𝑦 2 + √(𝑥 − (−𝑐)2 + 𝑦 2 = 2𝑎
Remember that the coordinate for F1 is (-c, 0) and F2 is (c, 0)
√(𝑥 − 𝑐)2 + 𝑦 2 + √(𝑥 + 𝑐)2 + 𝑦 2 = 2𝑎
The second addend moves to the other side of the equation and is
squared:
2
2
2
2
(√(𝑥 − 𝑐 ) + 𝑦 2 ) = (2𝑎 − √(𝑥 + 𝑐 ) + 𝑦 2 )
Each side of the equation is developed and we then simplify the similar
terms:
(𝑥 − 𝑐 )2 + 𝑦 2 = 4𝑎2 − 4𝑎 √(𝑥 + 𝑐 )2 + 𝑦 2 + (𝑥 + 𝑐 )2 + 𝑦 2
𝑥 2 − 2𝑐𝑥 + 𝑐 2 = 4𝑎2 − 4𝑎 √(𝑥 + 𝑐)2 + 𝑦 2 + 𝑥 2 − 2𝑐𝑥 + 𝑐 2
By grouping similar terms and factoring using a common factor, we
have:
−4(𝑐𝑥 + 𝑎2 ) = −4𝑎√(𝑥 + 𝑐)2 + 𝑦 2
2
(𝑐𝑥 + 𝑎 ) = (𝑎√(𝑥 + 𝑐)2 + 𝑦 2 )
2 2
By developing special products we have:
𝑐 2 𝑥 2 + 2𝑎2 𝑐 2 𝑥 2 + 𝑎4 = 𝑎2 𝑥 2 + 2𝑎2 𝑥𝑐 + 𝑎2 𝑐 2 + 𝑎2 𝑦 2
𝑎4 − 𝑎2 𝑐 2 = −𝑐 2 𝑥 2 + 𝑎2 𝑦 2 + 𝑎2 𝑥 2
Factoring 𝑎2 on one side and 𝑥 2 the other, we have:
𝑎2 (𝑎2 − 𝑐 2 ) = 𝑥 2 (𝑎2 − 𝑐 2 ) + 𝑎2 𝑦 2
In accordance with Pythagoras's theorem, 𝑎2 − 𝑐 2 = 𝑏 2 , by replacing, we
have:
𝑎2 𝑏 2 = 𝑥 2 𝑏 2 + 𝑎2 𝑦 2
By dividing everything between 𝑎2 𝑏 2 , we have:
𝑎 2 𝑏2
𝑥2 𝑏2
𝑎2 𝑦2
𝑎2 𝑏
𝑎2 𝑏
𝑎 2 𝑏2
=
2
1=
+
2
𝑥2 𝑦2
+
𝑎2 𝑏 2
The latter will be the ellipse equation, whose focus are on the x axis and
equidistant from the origin. Now, to find who will be 𝑎2 and 𝑏 2 , we must
remember that the length of the major axis will be 2 and of the minor
axis, 2b.
Ellipse equation with foci on the y axis
In a similar way to the previous, we can deduce that the equation of the
ellipse, whose foci are on the y axis, will be:
𝑦2 𝑥2
1= 2+ 2
𝑎
𝑏
We suggest you watch the following link, where you can see a step by
step demonstration. https://www.youtube.com/watch?v=OUZ7hUzzYMg
Now we will study a concrete case where we can find the equation under
specific conditions:
1. Find the equation of the ellipse whose major axis is located on the
x axis and measures 8 cm, while its minor axis measures 4 cm.
Graph it on the Cartesian plane.
2𝑎 = 8 → 𝑎 = 4
2𝑎 = 4 → 𝑎 = 2 Therefore, the ellipse equation will be:
𝑥2 𝑦2
1= 2+ 2
4
2
Thus, the graphical representation will be:
Image No. 11
Image designed by the author
Ellipse equation with foci anywhere on the Cartesian Plane.
In the same way as was done for the equations with foci on the axes,
we can see that the equation for an ellipse with foci at any point of the
plane will be:
If the major axis is horizontal:
(𝑥 − ℎ)2
𝑎2
+
(𝑦 − 𝑘)2
𝑏2
=1
If the major axis is vertical:
(𝑥 − ℎ)2
𝑏2
+
(𝑦 − 𝑘)2
𝑎2
=1
The coordinates (ℎ, 𝑘) correspond to the center of the ellipse.
In accordance with image No. 7, previewed before, the foci are found
on the major axis, c units from the center, taking into account
Pythagoras's theorem: 𝑐 2 = 𝑎2 − 𝑏 2
Examples:
1. Graphically represent and determine the coordinates of the foci,
the vertices, and the eccentricity of the following ellipse.
2
𝑥 + 2𝑦 2 − 2𝑥 + 8𝑦 + 5 = 0
For this type of equation you must complete the square, for both for x
and y.
(𝑥 2 − 2𝑥 + 1) − 1 + 2(𝑦 2 + 4𝑦 + 4) − 4 + 5 = 0
(𝑥 − 1)2 + 2(𝑦 + 2)2 = 1 + 8 − 5
By dividing everything between 4, we have:
(𝑥 − 1)2 (𝑦 + 2)2
+
=1
4
2
Since the major axis is horizontal, we have: 𝑎2 = 4 → 𝑎 = 2 On the other
hand,
𝑏 2 = 2 → 𝑏 = √2
With the equation we can deduce that the coordinates of the center of
the ellipse are: (1, -2), therefore, to find the major axis, it is known that
we must add 2 units to the left and 2 to the right.
Therefore, the coordinates of the vertices will be: (-1,-2) (3,-2)
Thus, the graphic representation of this ellipse will be:
Image No. 12
Image designed by the author
Learning activity
Questions 1 to 4 are answered according to the following
equation:
1=
𝑥2 𝑦2
+
16 25
For point 1, answer true or false:
1. The major axis is horizontal _________
2. The vertices of an ellipse are the points where the focal axis
intersects the ellipse. Taking this into account, the coordinates of
the vertices of this ellipse will be:
a) (0,5) and (0,-5)
b) (5,0) and (-5,0)
c) (5,5) and (0,-5)
d) (0,5) and (0,0)
Fill in the missing number
3. The length of the major axis is __________ while the minor axis is
______
4. The foci of the ellipse are located in:
a) (-3,3) and (0,-3)
b) (3,0) and (0,-3)
c) (0,3) and (0,-3)
d) (0,3) and (3,-3)
Abstract
Evaluation
1. The major axis of an ellipse is the segment that joins the vertices. For the
equation 3𝑥2 + 2𝑥2 = 6, it is correct to say that the major axis measures:
a) 2√3
b) √3
c) 3
d) 1.5
2. The graphical representation of the ellipse of point 1 is:
a)
c)
b)
d)
a) If the minor axis of an ellipse is horizontal, it implies that its ____________
will be located on the ____________ axis.
b) If an Ellipse has its foci in the coordinates (-2.3) and (1,3), the major axis will
be
Questions 4, 5 and 6 are answered according to the following equation: 3𝑥 2 + 𝑦 2 −
24𝑥 + 39 = 0
3. When completing the square for 𝑥 and 𝑥, the result will be:
a) 𝑥 2 + (𝑦 − 3)2 = 1
b) (𝑥 − 4)2 + (𝑦 − 39)2 = 1
c) 3(𝑥 − 4)2 + 𝑦 2 = 9
d) (𝑥 + 4)2 + (𝑦 + 39)2 = 1
Answer true or false:
4. The length of the major axis of the ellipse is 6 _________
5. The minor axis is vertical: _____________
6. Fill in the missing word:
Assignment/Homework
1.
a)
b)
c)
d)
Find the ellipse equation using the following information:
C(0,0), F (2,0), vertex (3,0)
C(0,0), F (0,4), vertex (0,5)
C(1,-1), F (1,2), vertex (1,4)
C(-3,2), F (-1,2), vertex (2,2)
2. Graphically represent and determine the coordinates of the foci, the
vertices, and the eccentricity of the following ellipses:
a) 𝑥 2 + 4𝑦 2 = 16
𝑦2
𝑥2
b) + = 1
25
9
2
c) 𝑥 + 3𝑦 2 − 6𝑥 + 6𝑦 = 0
Bibliography
1. Cano, L. (2015). Definición de grandes cónicas. Recuperado el 12
de
04
de
2016,
de
http://www.monografias.com/trabajos82/definicion-grandesconicas/definicion-grandes-conicas2.shtml
2. Garzón, A. (2013). Propuesta didáctica para la enseñanza de las
propiedades de reflexión de las cónicas por medio de la
metodología de resolución de problemas. La Elipse. (s.f.).
Recuperado
el
01
de
04
de
2016,
de
http://quiz.uprm.edu/pc_cb/elipse/elipse_right.xhtml