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EQUATION AND IMPORTANT
CHARACTERISTICS OF THE DIFFERENT
TYPES OF CONIC SECTIONS
for Pre Calculus
Senior High School (STEM)
Quarter 1 / Week 5
1
FOREWORD
This Self-Learning Kit (SLK) will serve as a guide in studying
the subject Area Pre-Calculus. Guided by the Most Essential
Learning Competencies (MELC), it will be used as an aid in
learning the concepts of conic sections and systems of
nonlinear equation.
Moreover, it gives focus on the knowledge,
understanding, skills , and attitudes that need to be
demonstrated in this lesson.
What happened (Let’s Do It)
This section contains pre-activities like review of the prior
knowledge of the different types of conic sections.
What I Need to Know (Discussion)
This section contains discussion on how to recognize the
equation and important characteristics of the different types
of conic sections wherein students will apply their
mathematical concepts learned.
What I have Learned (Evaluation/Post-Test)
The exercises contained in this section enhanced
student’s comprehension and mathematical skills. These
serve as a diagnostic tool to identify student’s weaknesses
and strengths.
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OBJECTIVES:
K. Identify the important characteristics of the equation of
the different types of conic sections.
S. Classify equations of the conic sections into circle,
parabola, hyperbola, or ellipse.
A. Appreciate the importance of the characteristics of the
different types of conic section to real life situation.
LEARNING COMPETENCY:
Recognize the equation and important characteristics of the
different types of conic sections (STEM_PC11AG – Ie – 1).
I. What Happened
PRE-TEST:
A. Classify the following equations whether circle, parabola, ellipse or
hyperbola. Answer in your activity sheet/notebook.
______ 1. 3x2 – 12x + 3y2 = 2
______ 2. y = x2 – 4
______ 3. 3x2 – 9x + 2y2 +10y – 6 =0
______ 4. 4y2 – 10y - 3x2 = 12
______ 5. X = 2y2 – 3y + 10
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II. What You Need to Know
In this lesson, we will discuss on how
to recognize the equatio In this
lesson, we will discuss on how to
recognize the equation and also
know the important characteristics of
the different types of conic section.
DISCUSSION:
A. Recognizing the Equation
The equation of a circle is written in a standard form
Ax2 + Ay2 – Cx + Dy +E = 0,
That is, the coefficients of x2 and y2 are the same. However, it does not follow
that if the coefficients of x2 and y2 are the same, the graph is a circle.
General Equation
Standard Equation
1
3
Graph
A.
2x2 + 2y2 – 2x + 6y + 5 = 0
(x- 2)2 + (y +2)2 = 0
point
B.
x2 + y2 – 6x + 8y + 50 = 0
(x - 3)2 + (y - 4)2 = -25
empty set
For a circle with equation (x – h)2 + (y – k)2 = r2, we have r2 > 0. This is not
the case for the standard equations of (A) and (B).
4
In (A), because the sum of two squares can only be 0 if and only if each
1
3
square is 0, it follows that x− 2= 0 and y + 2 = 0. The graph is thus the single point
1
3
(2, −2).
In (B), no real values of x and y can make the nonnegative left side equal
to the negative right side. The graph is then the empty set.
Example: General Form to Standard form.
2x 2 + 2 y 2 − 2x + 6 y + 5 = 0
General Form
(2 x − 2 x) (2 y + 6 y ) − 5
+
=
2
2
2
−5
( x 2 − x) + ( y 2 + 3 y ) =
2
1
9
−5 1 9
(x 2 − x + ) + ( y 2 + 3y + ) =
+ +
4
4
2 4 4
1
9
− 10 + 1 + 9
(x 2 − x + ) + ( y 2 + 3y + ) =
4
4
4
1
9
0
(x 2 − x + ) + ( y 2 + 3y + ) =
4
4
4
1
3
(x − )2 + ( y + )2 = 0
Standard Form
2
2
2
2
B. Important characteristics of conic section
Let us recall the general form of the equations of the other conic sections.
We may write the equations of conic sections we discussed in the general form
Ax2 + By2 + Cx + Dy + E = 0.
Some terms may vanish, depending on the kind of conic section.
(1) Circle: both x2 and y2 appear, and their coe๏ฌƒcients are the same
Ax2 + Ay2 + Cx + Dy + E = 0
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Example: 18x2 + 18y2 −24x + 48y−5 = 0 Degenerate cases: a point, and the
empty set
(2) Parabola: exactly one of x2 or y2 appears
Ax2 + Cx + Dy + E = 0 (D 6= 0, opens upward or downward)
By2 + Cx + Dy + E = 0 (C 6= 0, opens to the right or left)
Examples:
3x2 −12x + 2y + 26 = 0 (opens downward)
−2y2 + 3x + 12y−15 = 0 (opens to the right)
(3) Ellipse: both x2 and y2 appear, and their coe๏ฌƒcients A and B have the same
sign and are unequal
Examples:
2x2 + 5y2 + 8x−10y−7 = 0 (horizontal major axis)
4x2 + y2 −16x−6y + 21 = 0 (vertical major axis)
If A = B, we will classify the conic as a circle, instead of an ellipse.
Degenerate cases: a point, and the empty set
(4) Hyperbola: both x2 and y2 appear, and their coe๏ฌƒcients A and B have
different signs
Examples:
5x2 −3y2 −20x−18y−22 = 0 (horizontal transverse axis)
−4x2 + y2 + 24x + 4y−36 = 0 (vertical transverse axis)
Degenerate case: two intersecting lines
(4) Hyperbola: both x2 and y2 appear, and their coe๏ฌƒcients A and B have
different signs
Examples:
5x2 −3y2 −20x−18y−22 = 0 (horizontal transverse axis)
−4x2 + y2 + 24x + 4y−36 = 0 (vertical transverse axis)
Degenerate case: two intersecting lines
The following examples will show the possible degenerate conic (a point,
two intersecting lines, or the empty set) as the graph of an equation following a
similar pattern as the non-degenerate cases.
(๐‘ฅ−2)2
(1) 4x2 + 9y2 −16x + 18y + 25 = 0
(๐‘ฆ + 1)2
=⇒ 32 + 22 = 0
=⇒ one point: (2, −1)
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(๐‘ฅ−2)2
(๐‘ฆ + 1)2
(2) 4x2 + 9y2 −16x + 18y + 61 = 0
=⇒ 32 + 22
=⇒ empty set
(3) 4x2 −9y2 −16x−18y + 7 = 0
=⇒
(๐‘ฅ−2)2
32
-
(๐‘ฆ + 1)2
22
= -1
=0
2
=⇒ two lines: y + 1 = ± 3(x−2)
A Note on Identifying a Conic Section by Its General Equation
It is only after transforming a given general equation to standard
form that we can identify its graph either as one of the degenerate
conic sections (a point, two intersecting lines, or the empty set) or
as one of the non-degenerate conic sections (circle, parabola,
ellipse, or hyperbola).
1. Parabola
The parabola is a conic section, the intersection of a right circular conical
surface and a plane parallel to a generating straight line of that surface. The
equation for a parabola is
y=a(x−b)2+c or x =a(y−b)2+c
2. Circles and Ellipses
The equation of a circle with center at (a, b) and radius r units is
(x−a) 2 + (y−b) 2 = r2
An ellipse is the figure consisting of all points in the plane whose coordinates
satisfy the equation
๐ฑ๐Ÿ
๐ฒ๐Ÿ
+ =1
๐š๐Ÿ ๐› ๐Ÿ
If the ellipse has its center at (m, n) the equation could be written as
(๐ฑ−๐ฆ)๐Ÿ
๐š๐Ÿ
+
(๐ฒ−๐ง)๐Ÿ
๐›๐Ÿ
7
=1
3. Hyperbolas
A hyperbola is a curve, specifically a smooth curve that lies in a plane,
which can be defined either by its geometric properties or by the kinds of
equations for which it is the solution set. A hyperbola has two pieces, called
connected components or branches, which are mirror images of each other and
resembling two infinite bows.
The equation of a hyperbola with a center at (m, n) is
(x−m)2/a2 − (y−n)2/b2 = 1
(๐ฑ−๐ฆ)๐Ÿ
๐š๐Ÿ
-
(๐ฒ−๐ง)๐Ÿ
๐›๐Ÿ
=1
III. What Have I Learned
POST TEST:
The graphs of the following equations are (nondegenerate) conic sections.
Identify the conic section. Write your answer in your activity sheet/notebook.
(1) 5x2 −3y2 + 10x−12y = 22
(2) 2y2 −5x−12y = 17
(3) 3x2 + 3y2 + 42x−12y = −154
(4) 3x2 + 6x + 4y = 18
(5) 7x2 + 3y2 −14x + 12y = −14
(6) −4x2 + 3y2 + 24x−12y = 36
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DEPARTMENT OF EDUCATION
SCHOOLS DIVISION OF NEGROS ORIENTAL
SENEN PRISCILLO P. PAULIN, CESO V
Schools Division Superintendent
FAY C. LUAREZ, TM, Ed.D., Ph.D.
OIC - Assistant Schools Division Superintendent
Acting CID Chief
ADOLF P. AGUILAR
OIC - Assistant Schools Division Superintendent
NILITA L. RAGAY, Ed.D.
OIC - Assistant Schools Division Superintendent
ROSELA R. ABIERA
Education Program Supervisor – (LRMS)
ARNOLD R. JUNGCO
Education Program Supervisor – (SCIENCE & MATH)
MARICEL S. RASID
Librarian II (LRMDS)
ELMAR L. CABRERA
PDO II (LRMDS)
REONELL K. SELARDE
Writer
LITTIE BETH S. BERNADEZ
Lay-out Artist
_________________________________
ALPHA QA TEAM
LITTIE BETH S. BERNADEZ
MERCYDITHA D. ENOLPE
RONALD TOLENTINO
DIDITH T. YAP
BETA QA TEAM
ELIZABETH A. ALAP-AP
EPIFANIA Q. CUEVAS
NIDA BARBARA S. SUASIN
VRENDIE P. SYGACO
MELBA S. TUMARONG
HANNAHLY I. UMALI
DISCLAIMER
The information, activities and assessments used in this material are designed to provide accessible
learning modality to the teachers and learners of the Division of Negros Oriental. The contents of this module are
carefully researched, chosen, and evaluated to comply with the set learning competencies. The writers and
evaluator were clearly instructed to give credits to information and illustrations used to substantiate this material.
All content is subject to copyright and may not be reproduced in any form without expressed written consent from
the division.
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REFERENCES
Bacani, Jerico B., Eden, Richard B., Estrada, Glenn Rey A., Francisco, Flordeliza F., Vidallo,
Mark Anthony J., Department of Education, Ground Floor Bonifacio Building, Dep.Ed
Complex Meralco Avenue, Pasig City, Philippines 1600, 2016, Page 60-62.
https://www.cliffsnotes.com/study-guides/algebra/algebra-ii/conic-sections/the-fourconic-sections
https://www.mathplanet.com/education/algebra-2/conic-sections/equations-ofconic-sections
https://www.dummies.com/education/math/calculus/how-to-identify-the-four-conicsections-in-equation-form/
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SYNOPSIS AND ABOUT THE AUTHOR
SYNOPSIS
The Self Learning Kit (SLK) is
developed to prepare students
explore mathematical problems
involving conic sections.
Students are provided with
practice activities, examples
and assessment to test their skills
and knowledge, sharpen their
critical thinking skills and lead
them to develop wise decision
making.
ANSWERS KEY
Pre activities / Pre-test
A.
1. Circle
2. Parabola
3. Ellipse
4. Hyperbola
5. Parabola
Evaluation/ Post test
1. Hyperbola
2. Parabola
3. Circle
4. Parabola
5. Ellipse
6. Hyperbola
AUTHOR
Reonell K. Selarde is a graduate of Bachelor of Science in
Industrial Engineering at Foundation University, Dumaguete
City. Also finished his Continuing Professional Education at
Presbyterian Theological College Inc., Dumaguete City. He is
a Grade 12 teacher-adviser Maria Macahig Memorial High
School, Siaton, Negros oriental and has been serving the
Department of Education since 2018.
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