Salvacion National High

School Grade Level 11

School
Lesson Geraldine L. Dumaran-
Teacher Learning Area PRE-CALCULUS
Exemplar Libarra
Teaching Dates and 08/12/13/2024
Quarter 1st
Time 3:00-5:00PM

I. OBJECTIVES
The learner demonstrates understanding of key concepts of conic sections and
A. Content Standards systems of nonlinear equations the key concepts of series and mathematical
induction and the Binomial Theorem.
The learner can model situations appropriately and solve problems accurately
using conic sections and systems of nonlinear equations and shall be able to
B. Performance Standards
keenly observe and investigate patterns and formulate appropriate mathematical
statements.
The learners,
1. Define an ellipse.
C. Learning Competencies
2. Determine the standard form of equation of an ellipse.

II. CONTENT ELLIPSE

III. LEARNING RESOURCES
Agot M. Pre-Calculus. Department of Education – Division of Cagayan de Oro
City
Pre-Calculus. Teacher’s Guide (2016). DepEd-BLR. Sunshine Interlinks
A. References Publishing House, Inc. Quezon City, Philippines.
Pre-Calculus. Teacher’s Guide (2016). DepEd-BLR. Sunshine Interlinks
Publishing House, Inc. Quezon City, Philippines.
Curriculum Guide
B. Other Learning
Leaning Modules, Activity Sheets, Laptop, and TV
Resources
IV. PROCEDURES
A. Reviewing the previous The learners will give examples of ellipse that they can see in their everyday
lesson/Presenting the living.
new lesson Example: the orbits of the planets around the sun.
Base from those samples, they will define ellipse.
Learners will define ellipse base on its parts and characteristics that is modelled
by the teacher.
 Let F1 and F2 be two distinct points. The set of
B. Establishing a purpose all points P, whose distances from F1 and from
for the lesson F2 add up to a certain constant, is called an
ellipse. The points F1 and F2 are called the foci
of the ellipse.

C. Presenting Consider the points F1(−3, 0) and F2(3, 0), as shown in Figure 1.22. What is
examples/instances of the sum of the distances of A(4, 2.4) from F1 and from F2? How about the
the new lesson sum of the distances of B(and C(0, −4)) from F1 and from F2?
AF1 + AF2 = 7.4 + 2.6 = 10
BF1 + BF2 = 3.8 + 6.2 = 10
CF1 + CF2 = 5 + 5 = 10
There are another points P such that PF1 + PF2 = 10. The collection of all such
points forms a shape called an ellipse.

Figure 1.22 Figure 1.23

Let F1 and F2 be two distinct points. The set of all points P, whose distances
from F1 and from F2 add up to a certain constant, is called an ellipse. The
points F1 and F2 are called the foci of the ellipse.
Given are two points on the x-axis, F1( -c, 0) and F2(c, 0), the foci, both c units
away from their center (0, 0). See Figure 1.23. Let P(x, y) be a point on the
ellipse. Let the common sum of the distances be 2a (the coefficient 2 will make
computations simpler). Thus, we have PF1 + PF2 = 2a

P F1=2 a−P F2
 √(x +c )2+ y 2=2 a− √( x−c)2 + y 2
x + 2 cx+ c + y =4 a −4 a √ ( x−c ) + y + x −2cx + c + y
2 2 2 2 2 2 2 2 2

a √ ( x−c ) + y =a −cx
2 2 2

a 2 [ x 2−2 cx +c 2 + y 2 ]=a 4−2 a 2 cx +c 2 x2

( a 2−c 2 ) x2 + a2 y 2=a4 −a2 c 2=a 2 ¿)
b x + ¿ a y =a b by letting b=√ (a −c ) , so a>b
2 2 2 2 2 2 2 2

2 2
x y
2
+ 2
=1
a b

D. Discussing new An Ellipse is a set of all points in a plane, the sum whose distances from two
concepts and practicing fixed points is constant. The fixed points are called foci.
new skills #1 Learners will write the equation of the ellipse in

This is how to convert general form of ellipse to its standard form and vice versa.
Example 1. Convert the following general equation to standard form.
2 2
9 x + 8 y =288
Solution:
2 2
9 x + 8 y =288
2 2
9 x 8 y 288
+ =
288 288 288
2 2
x y
+ =1
32 36
2 2
x y
The standard form is + =1
32 36
Example 2. Convert the following standard form to general form:
2
( y −2)
+¿¿
25
Solution:
2
( y −2)
+¿¿
25
225 ¿
2 2
9( y−2) + 25( x−3) =225
9( y ¿¿ 2−4 y +4)+25 ¿ ¿
2 2
9 y −36 y +36 +25 x −150 x +225=225
2 2
9 y −36 y +36 +25 x −150 x +225−225=0
2 2
25 x + 9 y −150 x−36 y +36=0
The general form is 25 x 2+ 9 y 2−150 x−36 y +36=0


Learners will graph ellipse in the rectangle coordinate plane.
When we let b=a 2−c 2, we assumed a > c. To see why this is

true, look at
∆ PF1F2 in Figure 1.23. By the Triangle Inequality, PF1 + PF2 > F1F2,
which implies 2a > 2c, so a > c.
We collect here the features of the graph of an ellipse with
standard equation
2 2
x y
2
+ 2 =1,where a>b. Let c= √ (a2−b2) .
a b

Figure 1.24

 center : origin (0, 0)

 foci : F1(−c, 0) and F2(c, 0)
o Each focus is c units away from the center.
o For any point on the ellipse, the sum of its distances
from the foci is 2a.
  vertices: V1(-1,0) and V2 (1,0)
o The vertices are points on the ellipse, collinear with the
center and foci.
o If y = 0, then x = ±a. Each vertex is a unit away from the
center. The segment V1V2 is called the major axis. Its
length is 2a. It divides the ellipse into two congruent
parts.
 covertices: W1(0, −b) and W2(0, b)
o The segment through the center, perpendicular to the
major axis, is the minor axis. It meets the ellipse at the
covertices. It divides the ellipse into two congruent
parts.
o If x = 0, then y = ±b. Each covertex is b units away from
the center.
o The minor axis W1W2 is 2b units long. Since a > b, the
major axis is longer than the minor axis.

E. Discussing new In the standard equation, if the x-part has the bigger denominator, the ellipse is
concepts and practicing horizontal. If the y-part has the bigger denominator, the ellipse is vertical
new skills #2
Let us find out if you really understand the discussed concept by answering
these exercises.

1. Convert the following general form to standard form of an ellipse.

a. 16 x 2+ 4 y 2−32 x +16 y−32=0 c. 9 x 2+ 4 y 2 −72 x−24 y+ 144=0

F. Developing mastery b. 4 x 2+ 9 y 2+ 48 x+ 72 y +144=0 d. 49 x 2 +9 y 2=441

2. Convert the following standard form to general form of an ellipse .

a. ¿ ¿ c. ¿ ¿

b¿ ¿ d. . ¿ ¿

Give the coordinates of the center, foci, vertices, and covertices of the ellipse
with the given equation. Sketch the graph and include these points.
G. Finding practical
applications of concepts 1.
and skills in daily living
2.

H. Making generalization Let me check your knowledge by filling the blanks with a correct
about the lesson symbols/letter or terms in order to complete the statement/s.
1. An ellipse is the set of all points in the plane for which the __________ of the
distances from two fixed points F1 and F2 is constant. The points F1 and F2 are
called the _______ of the ellipse.
2 2
x y
2. The graph of equation 2
+ 2 =1 with a>b is an ellipse with vertices (__,__)
a b
2 2
x y
and (__,__) and foci (± c ,0 ¿ ,where c = ________. So the graph of 2 + 2 =1
5 4
is an ellipse with vertices (_,_) and (_,__) and foci (_,__) and (__,__).
2 2
x y
3. The graph of the equation 2
+ 2 =1with a>b>0 is an ellipse with vertices
b a
2 2
x y
(__,___ and (__,__) and foci (0,± c ¿ where c=___. So the graph of 2 + 2 =1 is
4 5
an ellipse with vertices (__,__) and (__,__) and foci (__,__) and (__,__).
4. Label the vertices and foci on the graphs given for the ellipses:
 2 2 2 2
x y x y
a. 2 + 2 b. 2 + 2 =1
5 4 4 5

I. Evaluating learning Directions: Choose the best answer. Encircle the letter of your choice.
For numbers 1-4, refer to the equation 25 x 2−200 x+ 16 y 2−160 y=800 , and
give what is being asked.
1. Standard form
( x−4 )2 ( y−5 )2
a. +¿ =1 c.
64 100
( x+ 4 )2 ( y+5 )2
+ =1
64 100
( x−4 )2 ( y−5 )2
b. +¿ =1 d.
100 64
( x+ 4 )2 ( y+5 )2
+ =1
100 64
2. Center
a. (4,5) c. (-5,-4)
b. (-4,-5) d. (5,4)
3. Foci
a. (4 ,−1)( 4 , 11) c. (−4 , 5)(12, 5)
b. (4 ,−5)(4 , 15) d. (−4 , 1)(−4 ,−11)
4. Vertices
a. (4 ,−1)( 4 , 11)
b. (−4 , 5)(12, 5)
c. (4 ,−5)( 4 , 15)
d. (−4 , 5)(−4 ,−15)
5. Covertices
a. (−11,−5)(4 ,−5)
b. (−4 , 5)(12, 5)
c. (4 ,−5)( 4 , 15)
d. (−4 , 5)(12, 5)
6. Graph
a. Yellow
b. Pink
c. Blue
d. Green
For numbers 7 and 8, refer to the given situation.
A 40ft wide tunnel has the shape of a semi ellipse that is 5ft high a distance of
2ft from each end.
7. How high is the tunnel at its center?
a. 31.35 ft c. 11.47 ft
b. 25.06 ft d. 25.25 ft
8. Is it possible for a 7ft tall and 5 ft wide vehicle to pass through the
tunnel?
a. Yes
b. No
For numbers 9 and 10, refer to the given situation.
A big room is constructed so that the ceiling is a dome that is semielliptical in
shape. If a person stands at one focus and speaks, the sound that is made
bounces off the ceiling and gets reflected to the other focus. Thus, if two people
stand at the foci (ignoring their heights), they will be able to hear each other.
9. If the room is 34 m long and 8 m high, how far from the center should
each of two people stand if they would like to whisper back and forth
and hear each other?
a. √ 353 m c. 225 m
b. 15 m d. 30 m
10. Is it possible for the two people who are 30 meters away from each
other to hear their whispers?
 a. Yes
b. No

J. Additional activities Read in advance about hyperbola.

V. REMARKS
VI. REFLECTION
A. Attendance
B. Index of Proficiency
C. Index of Mastery
D. Most mastered item
E. Least mastered item
F. Observer Name and
Signature
G. Date of Observation

Prepared by: Checked by:

GERALDINE D. LIBARRA PERPETUA L. BACUEL

T-II, Pre-Cal Subject Teacher HT-III, MATH Department