SENIOR

Precalculus HIGH
SCHOOL

Module
Circle with Center at (h, k)
3
Quarter 1
Precalculus
Quarter 1 – Module 3: Circle with Center at (h, k)
First Edition, 2020

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Development Team of the Module

Writer: Erwin C. Lugtu
Editor: Nenet M. Peñaranda
Reviewers: Emma G. Gonzalvo, Raffy D. Maga, Annie R. Ascotia, Ron Robert Pecaña
Cerina V. Galoy (Technical)
Illustrator:
Layout Artist:
Management Team: Ma. Evalou Concepcion A. Agustin
OIC-Schools Division Superintendent
Dr. Aurelio G. Alfonso
OIC-Assistant Schools Division Superintendent
Dr. Victor M. Javeña
Chief, School Governance and Operations Division and
OIC-Chief, Curriculum Implementation Division
Education Program Supervisors

Librada L. Agon, Ed. D., EPP/TLE

Liza A. Alvarez, Science
Bernard R. Balitao. Araling Panlipunan
Joselito E. Calios, English
Norlyn D. Conde Ed. D., MAPEH
Wilma Q. Del Rosario, LRMS
Ma. Teresita E. Herrera, Ed. D., Filipino
Perlita M. Ignacio, Ph. D. ESP/SPED
Dulce O. Santos, Ed. D., Kinder/ MTB
Teresita P. Tagulao, Ed. D., Mathematics

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 SENIOR
Precalculus HIGH
SCHOOL

Module

3
Quarter 1

Circle
with Center at (h, k)
Introductory Message

For the facilitator:

Welcome to the Precalculus for Senior High School Module on Circle with Center at
(h, k)!

This module was collaboratively designed, developed and reviewed by educators

from Schools Division Office of Pasig City headed by its Officer-In-Charge Schools
Division Superintendent, Ma. Evalou Concepcion A. Agustin in partnership with
the Local Government of Pasig through its mayor, Honorable Victor Ma. Regis N.
Sotto.
The writers utilized the standards set by the K to 12 Curriculum using the Most
Essential Learning Competencies (MELC) while overcoming their personal, social,
and economic constraints in schooling.

This learning material hopes to engage the learners into guided and independent
learning activities at their own pace and time. Further, this also aims to help
learners acquire the needed 21st century skills especially the 5 Cs namely:
Communication, Collaboration, Creativity, Critical Thinking and Character while
taking into consideration their needs and circumstances.

In addition to the material in the main text, you will also see this box in the body of
the module:

Notes to the Teacher

This contains helpful tips or strategies
that will help you in guiding the learners.

As a facilitator you are expected to orient the learners on how to use this module.
You also need to keep track of the learners' progress while allowing them to
manage their own learning. Moreover, you are expected to encourage and assist the
learners as they do the tasks included in the module.
For the learner:

Welcome to the Precalculus Module on Circle with Center at (h, k)!

The hand is one of the most symbolized part of the human body. It is often used to
depict skill, action and purpose. Through our hands we may learn, create and
accomplish. Hence, the hand in this learning resource signifies that you as a
learner is capable and empowered to successfully achieve the relevant
competencies and skills at your own pace and time. Your academic success lies in
your own hands!

This module was designed to provide you with fun and meaningful opportunities
for guided and independent learning at your own pace and time. You will be
enabled to process the contents of the learning material while being an active
learner.

This module has the following parts and corresponding icons:

Expectation - These are what you will be able to know after completing the
lessons in the module

Pre-test - This will measure your prior knowledge and the concepts to be
mastered throughout the lesson.

Recap - This section will measure what learnings and skills that you
understand from the previous lesson.

Lesson- This section will discuss the topic for this module.

Activities - This is a set of activities you will perform.

Wrap Up- This section summarizes the concepts and applications of the
lessons.

Valuing-this part will check the integration of values in the learning

competency.

Post-test - This will measure how much you have learned from the entire
module.
 EXPECTATIONS

At the end of the learning episode, you are expected to:

1. graph a circle with center at (h, k) given an equation in center-radius

form.

PRETEST

Write the letter of the correct answer on your answer sheet.

1. Assume we pick a point (x, y) on the circle considering its center at (h,
k). What is the radius?
A. 𝑟 = √(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 C. 𝑟 = √(𝑥 + ℎ)2 − (𝑦 + 𝑘)2
B. 𝑟 = √(𝑥 − ℎ)2 − (𝑦 − 𝑘)2 D. 𝑟 = √(𝑥 + ℎ)2 + (𝑦 + 𝑘)2
2. What is the center-radius form equation of a circle of radius r with
center at (h, k)?
A. (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 C. 𝑟 = (𝑥 + ℎ)2 + (𝑦 + 𝑘)2
B. (𝑥 − ℎ)2 − (𝑦 − 𝑘)2 = 𝑟 2 D. 𝑟 = (𝑥 + ℎ)2 − (𝑦 + 𝑘)2
3. Which of the following is the radius of the circle with center at (3, 5)
touching point (-1, 2)?
A. 3 C. 5
B. 4 D. 6
4. What is the center-radius form equation of the circle given in item
number 3?
A. (𝑥 + 3)2 + (𝑦 + 5)2 = 25 C. (𝑥 − 3)2 + (𝑦 − 5)2 = 25
B. (𝑥 − 1)2 + (𝑦 + 2)2 = 25 D. (𝑥 + 1)2 + (𝑦 − 2)2 = 25
 5. Which of the following is the graph of the circle (𝑥 + 2)2 + (𝑦 − 1)2 = 16?
A. C.

B. D.

RECAP

We have discussed from the previous module that a circle consists of all
points that are at a fixed distance r, its radius, from a fixed point C, its
center.

Any point (x, y) on the circle with center (0, 0) and radius r must satisfy the
standard equation 𝒙𝟐 + 𝒚𝟐 = 𝒓𝟐 . This equation is called the center-radius
form equation of the circle with center (0, 0) and radius r, r>0.
To graph a circle given its center-radius form equation, consider the
following steps:

1. Plot the center given its coordinates

2. On the axes, plot the four points of the circle using the given r as
the distance from the center.
3. Connect the 4 points in Step 2 to form a circle
A circle whose center is at (0, 0) and has a radius of 1 unit has the standard
equation 𝑥 2 + 𝑦 2 = 1 and is called the unit circle.
To determine if the given point (x, y) is on the circle 𝑥 2 + 𝑦 2 = 𝑟 2 , substitute
the values of x and y and the sum must be equal to the value of 𝑟 2 .

To better check your understanding on the previous lesson, try to answer

this Review Exercise.

Review Exercise:
I. Identify the radius of the following:

1. 𝑥2 + 𝑦2 = 1
2. 𝑥2 + 𝑦2 − 9 = 0
3. Center at (0, 0) with a point on the circle (3, -5)
4. 𝑦 2 = 16 − 𝑥 2
II. Find the center-radius form equation and sketch the graph of the circle
with center at (0, 0) given the following radius.

1. r = 2
1
2. r = 3

LESSON

We start this lesson by defining again a circle. A circle is the locus of all
points in the plane having the same fixed positive distance r, called
radius, from a fixed point C, called the center. Suppose that a circle is
located in a Cartesian Plane so that its center C is (h, k) and its radius is r.
If a point 𝑃(𝑥, 𝑦) is on the circle, its distance from 𝐶(ℎ, 𝑘) has to be r units
(See Figure 3.1)
By the distance formula 𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 , x and y satisfy the
equation √(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 or (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 .

Conversely, any ordered pair (x, y) that satisfies this equation defines a point
P(x, y) that lies on the circle with center at (h, k) and radius r. Thus we have
the following result:

Center-Radius Form Equation of a Circle with Center at (h, k)

Any point (x, y) on the circle with center (h, k) and radius r must satisfy
the standard equation
(𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐 , 𝑟 > 0.

Example 1:

Find the center-radius or standard form equation for the circle of radius 4
with center at (-1, 3) and sketch its graph.

Solution:

From the given problem, we identified the values of r, h, and k as 4, -

1, and 3, respectively.

Substituting 4 for r, -1 for h, and 3 for k, in the equation (𝑥 − ℎ)2 +

(𝑦 − 𝑘)2 = 𝑟 2 , we get

[𝑥 − (−1)]2 + (𝑦 − 3)2 = 42

or
(𝒙 + 𝟏)𝟐 + (𝒚 − 𝟑)𝟐 = 𝟏𝟔

To graph the circle, consider the following steps:

Example 2:

Find the center-radius or standard form equation and sketch the graph of
the circle with center at (2, 5) tangent to the x-axis. (Recall that when a circle
is tangent to x-axis, that circle touches the x-axis at exactly one point. The
segment connecting this intersection point and the center of a circle is
perpendicular to the x-axis.)

Solution:

The center of the circle is at (2, 5). So, h = 2 and k = 5.

Since the circle is tangent to x-axis, the point of intersection of the

circle and x-axis is at (2, 0). The distance from C(2, 5) and (2, 0) is 5.
Therefore, r = 5.

Substituting 5 for r, 2 for h, and 5

for k, in the equation (𝑥 − ℎ)2 +
(𝑦 − 𝑘)2 = 𝑟 2 , we get

(𝑥 − 2)2 + (𝑦 − 5)2 = 52

or
(𝒙 − 𝟐)𝟐 + (𝒚 − 𝟓)𝟐 = 𝟐𝟓

Following the steps of graphing a

circle given above, the graph of
(𝑥 − 2)2 + (𝑦 − 5)2 = 25 is given in Figure 3.6.

Example 3:

Find the center-radius or standard form equation and sketch the graph of
the circle with center at (2, 5) tangent to the y-axis. (Recall that when a circle
is tangent to y-axis, that circle touches the y-axis at exactly one point. The
segment connecting this intersection point and the center of a circle is
perpendicular to the y-axis.)

Solution:

The center of the circle is at (2, 5). So, h = 2 and k = 5.

Since the circle is tangent to y-axis, the point of intersection of the

circle and y-axis is at (0, 5). The distance from C(2, 5) and (0, 5) is 2.
Therefore, r = 2.

Substituting 2 for r, 2 for h, and 5 for k, in the equation (𝑥 − ℎ)2 +

(𝑦 − 𝑘)2 = 𝑟 2 , we get

(𝑥 − 2)2 + (𝑦 − 5)2 = 22

or
(𝒙 − 𝟐)𝟐 + (𝒚 − 𝟓)𝟐 = 𝟒

Following the steps of graphing a circle given above, the graph of

(𝑥 − 2)2 + (𝑦 − 5)2 = 4 is given in Figure 3.7.

Example 4:
A circle has a diameter with endpoints A(-1, 5) and B(2, 1). Find its center-
radius form equation and sketch its graph.

Solution:

We let 𝑥1 = −1, 𝑦1 = 5, 𝑥2 = 2, 𝑎𝑛𝑑 𝑦2 = 1.

The diameter of the given circle is the distance between A and B.

Using the distance formula, we get

𝑑(𝐴𝐵) = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2

𝑑(𝐴𝐵) = √(2 − (−1))2 + (1 − 5)2 = √(3)2 + (−4)2 = √9 + 16 = √25 = 5

5
Since the radius is half of a diameter, then 𝑟 = .
2

It is also known that the midpoint of a diameter is the center of the

circle. We let the midpoint of A and B to be the coordinates of the center (h,
k). Then,
𝑥1 +𝑥2 −1+2 1
ℎ= = =
2 2 2
𝑦1 + 𝑦2 5 + 1 6
𝑘= = = =3
2 2 2
5 1
Substituting 2
for r, 2
for h, and 3 for k, in the equation (𝑥 − ℎ)2 +
(𝑦 − 𝑘)2 = 𝑟 2 , we get

1 2 2
5 2
(𝑥 − ) + (𝑦 − 3) = ( )
2 2
or
𝟏 𝟐 𝟐𝟓
(𝒙 − ) + (𝒚 − 𝟑)𝟐 =
𝟐 𝟒

Following the steps of graphing a circle given above, the graph of

1 2 25
(𝑥 − ) + (𝑦 − 3)2 = is given in Figure 3.8.
2 4
 ACTIVITIES

Complete the table:

Equation of Circle in
Center Radius Sketch of the Graph
Center-Radius Form

(𝑥 − 1)2 + (𝑦 + 3)2 = 16

(0, 4) 3
 WRAP–UP

From what we have discussed in this module, we learned the following:

1. A circle is the locus of all points in the plane having the same fixed
positive distance r, called radius, from a fixed point C, called the
center.
2. Given any point (x, y) on the circle with center (h, k) and radius r must
satisfy the standard equation (𝒙 − 𝒉)𝟐 + (𝒚 − 𝒌)𝟐 = 𝒓𝟐 , 𝑟 > 0. This
equation is the center-radius form equation of a circle with center
at (h, k)
3. This equation was derived from the distance formula.
4. To be able to graph a circle, the coordinates h and k of its center and
the radius r are considered.

VALUING

Your reflections in the last module discussed God as our origin. He loves us
equally. In this module, we learned from our discussion that the equation of
a circle, whatever the center is located, will be based from the length of its
radius. It says that the equation is (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2 .
Can you say now that God still loves us equally even if He is not in constant
place? We need to realize that God is everywhere, He can be anywhere. God
still loves us equally wherever He is. That’s the unconditional love of God to
us.

POSTTEST

I. Identify the radius of the following:

1. center at (3, -5) and a point on the circle (-7, 4)

 2. (𝑥 + 1)2 + 𝑦 2 = 12
3. diameter with endpoints (-5, -4) and (7, -3)
4. center at (4, 5) tangent to x-axis

5.

II. Write the center-radius form and sketch the graph of the circle with center at (h,
k) given the following conditions:

1. Center at (-2, 3) and radius 4

2. Diameter with endpoints (4, 2) and (-2, -6)
3. Center at (3, 1) tangent to y-axis
KEY TO CORRECTION
 REFERENCES

Books:
1. Department of Education-Bureau of Learning Resources (2016).
PRECALCULUS LEARNER’S MATERIAL.
2. Feliciano and Uy (1994). MODERN ANALYTIC GEOMETRY. Merriam
and Webster Bookstore, Inc. Metro Manila, Philippines
3. Munem and Yizze (2002). Precalculus: Functions and Graphs Seventh
Edition. Kendall/Hunt Publishing Company, Dubuque, Iowa 52002
4. Pelias, JG P (2016). PRE-CALCULUS. Rex Book Store, Inc. (RBSI).
Sampaloc, Manila
ACKNOWLEDGEMENT

EDISON P. CLET
Illustrator

ELINETTE B. DELA CRUZ

Project Development Officer II (LRMS)
Lay-out Artist

ERWIN C. LUGTU
Video/PowerPoint Presenter

NAME
Video/ PowerPoint Editor

NAME
Video/ PowerPoint Reviewer

ERWIN C. LUGTU
School Subject Coordinator

MONETTE P. VEGA
School Head

DIANA MARIE P. DAGLI

Public Schools District Supervisor