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Chapter 2: Circle
SSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
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Chapter 2: Circles Lecture 2: Introduction to Circles
Lecture 3: Form of a Circle
Lecture 4: General Form and
Standard Form of a Circle Lecture 5: Circles Determined by
Different Conditions Lecture 6: Tangent to a Circle
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Nice to Know:
TED Ed Video: Why
are Manholes Cover
Round? by Marc
Chamberland
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From TED Ed Video:
Keep your eyes open
and you just might come
across a rule of triangle
manhole.
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Nice to Know:
Why is πr2 the
area of a circle?
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Lecture 2: Introduction to Circle
SSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
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What is a Circle?
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Definition of Circle
A circle is a set of all points in a plane whose distances from a
fixed point is a constant.
),( yx
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The Center and the Radius
The fixed point is called the CENTER, and the distance
from the center to any point of the circle is referred to as the
RADIUS.
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The Derivation of Equation of a Circle
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Something to think about…If the radius is the distance from a
fixed point to any point on the circle, what formula we have
learned from junior high school can we use to derive the standard
equation of a circle?
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The Distance Formula:
212
2
12 yyxxd
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Form of Circle # 1:The Standard Form of an
Equation of a Circle with Radius rand Center at (h, k):
222rkyhx
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Something to think about…
What will happen to the standard equation of a
circle with radius r if the center is at the origin?
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Form of Circle # 2:The Standard Form of an
Equation of a Circle with Radius rand Center at the origin (0, 0):
222 ryx
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Something to think about…
What will happen to the standard equation of a circle with the center at the origin if the radius is
1?
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Form of Circle # 3:The Standard Form of an
Equation of a Circle with Radius 1and Center at the origin (0, 0):
122 yx
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Did you know?In addition, the standard form of an equation of the circle with radius 1 and
center at the origin is called UNIT CIRCLE and has the equation:
122 yx
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Form of Circle #4:The General Form of the
Equation of a Circle is:
022 FEyDxyx
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Lecture 3: Forms of a Circle
SSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
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Learning Expectation:
This lecture will discuss how to set up the graph of a circle and determine its
radius.
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Example 5:
Graph the circle with standard form of:
1622 yx
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Something to think about…
What do you know about radii of the
same circle?
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Example 6:Determine the general
equation of the circle whose center is (3, -1) and whose graph contains the point (7, -1). Also,
sketch the graph.
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Did you know?
We can convert standard equation of a circle to its
general form by expanding the binomials using the FOIL
method.
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Final Answer: Therefore, the general equation of the circle whose center is (3, -1) and whose
graph contains the point (7, -1) is:
062622 yxyx
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Example 7:
Find the general equation of the circle whose center is (2, 6)
and whose radius is 3. Also, please graph the circle.
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Final Answer:Thus, the general equation of the
circle whose center is (2, 6) and whose radius is 3:
03112422 yxyx
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Performance Task 2:
Please download, print
and answer the “Let’s
Practice 2.” Kindly work
independently.
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Lecture 4: Converting General Form to Standard Form of a Circle
SSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
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A Short Recap:
What is the standard form of the equation of the circle where the center is C (h, k)
and the radius is r?
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A Short Recap:
What is the general form of the equation
of a circle?
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Something to think about…
What benefit can we get from using standard equation instead of general equation of
a circle?
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Did you know?
The standard form of the circle is more convenient in the sense that we can easily identify the center and the
radius of a circle.
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Example 8:Change the general equation of a circle,
to standard form and determine the center and the radius. Also, please sketch
the graph.
06822 yxyx
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Did you know?
To convert general equation of a circle to its standard
equation, use the completing the square method.
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Step 1: Completing the Square
Group the equation according to its
variables.
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Step 2: Completing the Square
Ensure that the coefficients of x2 and
y2 are both 1.
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Step 3: Completing the Square
Take the half of the coefficient of x and y, squared it, and add it to both sides of
the equation.
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Step 4: Completing the Square
Factor by perfect square trinomial.
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Step 5: Completing the Square
Simplify the value of the radius (r2).
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Final Answer:
Thus, the standard form of the equation of the circle is
while the center is at (-4, 3)
and r = 5.
,253422 yx
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Example 9:Express the given general equation
below to its standard form:
0208422 yxyx
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Something to think about…
Is there an easiest way to convert the general form of a circle to its
standard form?
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Classroom Task 3:
By completing the square, derive the equations which we
can use to easily convert a circle in general form to its
standard form.
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Thus, center C (h, k) and radius r is equivalent to:
The center C (h, k) and radius r can be obtained using the following formula:
2
Dh
2
Ek
4
4222 FED
r
2
422 FEDr
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Something to think about…
The center C (h, k) is (2, 4). Also, note that the right side of the equation is zero. So, what can you conclude about this
circle?
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Point Circle or Degenerate CircleThus, point (2, 4) is the only point on a plane that satisfies the equation
and the radius is zero. This type of equation is referred as POINT CIRCLE or a DEGENERATE
CIRCLE.
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Example 10:
Express the general form of a circle below to its standard form:
04010622 yxyx
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Something to think about…
Revisit our previous examples, what have you
observed on the value of the
radius r2? What can you
conclude?
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Take Note:
The value of r2 in the standard equation of a circle is
always positive. If r2 is negative, then the solution
does not exist.
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Take Note:Note that the right side of the equation is
negative. This implies that there is no point in the plane that satisfies the equation
Therefore, the circle DOES NOT EXIST.
04010622 yxyx
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Something to think about…
Why do you think the value of radius r2 will
never ever be negative?
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To sum it up…
What conclusions can we make with respect to the
radius of circle?
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Conclusion Number 1:
Whenever the radius of a circle is a positive value,
the circle exists.
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Conclusion Number 2:
Whenever the radius of a circle is exactly equal to
zero, the circle is a point or degenerate circle.
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Conclusion Number 3:
Whenever the radius of a circle is a negative value, the circle does not exist.
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Performance Task 3:
Please download, print
and answer the “Let’s
Practice 3.” Kindly work
independently.
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Lecture 5: Circles Determined by Different Conditions
SSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
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What should you expect?This section illustrates how to establish the equation of a circle
given different conditions. Three of the possible cases are presented in
the succeeding examples.
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Example 11:
Determine the general equation of the circle which passes through the points P1 (-1, 2), P2 (0, 5), andP3 (2, 1). Also, please sketch the
graph.
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Something to think about…
Where can we find the given three
points on the circle?
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Method Number 1:
Method Number 1:Determining the GENERAL EQUATION given three points
on the circumference of the circle.
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Step 1:
Since we already have values
for (x, y), out of it we will
construct three (3) equations
which are of the form:
022 FEyDxyx
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Something to think about…What have you observed on our three equations? In order for us to find the values of D, E, and F, what
method we have learned from junior high school can we use?
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Step 2:Through the use of solving systems
of linear equation (elimination
method), we will combine the first
and second equation so we can eliminate variable(s).
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Step 3:Through the use of solving systems
of linear equation (elimination
method), we will combine the first
and third equation so we can eliminate variable(s).
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Step 4:
Combine the fourth and the fifth equations and solve for the value
of E.
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Step 5:
Substitute E = -6 to
the fourth equation to
obtain the value of D.
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Step 6:
To obtain the value of
F, we substitute also
E = -6.
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Step 7:
Lastly, we substitute
D = -2, E = -6, and
F = 5 to the general equation
of a circle.
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Method Number 2:
Method Number 2:Determining the STANDARD EQUATION given three points
on the circumference of the circle.
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Final Answer:
Thus, the general equation of the circle is
which contains the points P1 (-1,
2), P2 (0, 5), and P3 (2, 1).
056222 yxyx
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Example 12: Determine the equation of the
circle passing through P1 (4, 0), and
P2 (3, 5), with a linepassing through the
center. Also, please sketch the graph.
0723 yx
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Take Note: If is a line, by definition
is a collection of infinitely many points. Moreover, if passes
through the center, and remember by definition, the center is a fixed point, therefore the center of the circle is one of the infinitely
many points of .
0723 yx
0723 yx
0723 yx
0723 yx
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Also…If contains the fixed
point C (h, k), the coordinates of the center are solutions to the line
. Since its solutions are denoted by x and y, we will let C (h, k) be C
(x, y) as the center of the circle.
0723 yx
0723 yx
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Final Answer:Therefore, the equation of the circle passing through P1 (4, 0), and P2 (3, 5),
with a line passing through the center is:
0723 yx
084222 yxyx
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Example 13: Find the equation of the circle that circumscribe the triangle determined
by the lines x = 0, y = 0 andAlso, please sketch
the graph.
.02443 yx
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Something to think about…
When can we say a triangle is
circumscribed by a circle?
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Tell Me: “Which is Which?”From the two figures, which is
circumscribed circle and inscribed circle?
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Circumscribed CircleIn geometry, the circumscribed circle
or circumcircle of a polygon is a circle
which passes through all the vertices of the polygon. The center of this circle is
called circumcenter and its radius is
called the circumradius.
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Performance Task 4:
Please download, print
and answer the “Let’s
Practice 4.” Kindly work
independently.
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Lecture 6: Tangent to a Circle
SSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
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Did you know?
There are three (3)
saddest love stories in
Mathematics…
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The Painful Asymptote
There are people who
may get closer and
closer to one another,
but will never be
together.
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The Painful Parallel
You may encounter
potential people, bump onto
them, see them from afar, but
will never actually get to
know and meet them; even in
the longest time.
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The Painful Tangent
Some people are only
meant to meet one
another at one point in
their lives, but are
forever parted.
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Something to think about…
What are the three (3)
conditions that guarantee
a line is tangent to a
circle?
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Definition: Tangent to a Circle
A line in the plane of the circle that intersects the circle at exactly one
point is called tangent line. The point
of intersection is called the point of
tangency.
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The Tangent-Line Theorem:
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The Tangent-Line Theorem
“If a line is tangent to a
circle, then it is
perpendicular to the radius
at its outer endpoint.”
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The Tangent-Line Theorem:
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Something to think about…
If tangent is a line, what kind of function is a tangent line? Also, what’s its general
equation?
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Did you know?
A tangent is associated to a graph of a line. Thus, a tangent is
a linear function which has a
general equation of:
0 CByAx
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Did you know?
There are four main types of problems
concerning tangents to circles.
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The Four Main Types
1. Tangent at a Given Point2. Tangent in Prescribed Direction3. Inscribed Circle in a Triangle4. Tangents from a Point Outside
the Circle
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Tangent to a Circle:
Tangent at a Given PointSSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
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Example 14:
Given the equation of the circle
Prove that is a tangent to the circle. Sketch the graph.
.04514822 yxyx
02 yx
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Something to think about…
How can we prove thatis a tangent to the
circle ?04514822 yxyx
02 yx
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Analysis: If is a tangent line to the circle
, it touched the at exactly one point, and that point is what we
call the point of tangency. Moreover, since
is a tangent line, it is a collection of infinitely many points.
04514822 yxyx
02 yx
02 yx
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Analysis: If is tangent to the circle the point of
tangency PT (x, y) is one of the infinitely many
points of . Also, the point of tangency is on the circumference of the circle
. Hence, it is one of the infinitely many points in the circumference of the
circle .
04514822 yxyx
02 yx
02 yx
04514822 yxyx
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Analysis: Furthermore, if the tangent line and
the circumference of the circle .
both contain the point of tangency PT (x, y).
Therefore, the PT (x, y) is the solution to both the
tangent line and the circle .04514822 yxyx
02 yx
02 yx
04514822 yxyx
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Proving a Line is Tangent to a Circle We need to show that
touches the circle with equation.in a single point.
This single point is a common point of the tangent and the circle. Thus, it is the solution to the equation of the circle and the tangent
line.
04514822 yxyx
02 yx
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Take Note:
“If a line touches the
circle in a single point,
then it’s a tangent.”
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Take Note:
“If a line touches the
circle in two points,
then it’s a secant.”
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Take Note:
“If a line does not
touch the circle there
is no solution.”
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Final Answer:
Thus, the point of tangency is at (6,
3). Since there is only one solution, this shows that the line
just touches the circle in one place and therefore it is a tangent.
02 yx
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Example 15:
Find the equation of the tangent line to the circle
at the point (-2, 1). Sketch the graph.
01710622 yxyx
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Five Forms of Linear Equation:
The Slope-Intercept Form
bmxy
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Five Forms of Linear Equation:
The Point-Slope Form
)()( 11 xxmyy
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Five Forms of Linear Equation:
The Two-Point Form
)()( 1
12
121 xx
xx
yyyy
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Five Forms of Linear Equation:
The Intercept Form
0b
y
a
x
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Five Forms of Linear Equation:
The Normal Form
0sincos pyx
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The Tangent-Line Theorem
“If a line is tangent to a
circle, then it is
perpendicular to the radius
at its outer endpoint.”
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The Perpendicular Slope Theorem:
“If two lines are perpendicular, having respective slopes, m1 and
m2, the slope of the line is the negative reciprocal of the slope of
the other line:
."1
1
2m
m
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Final Answer:
Therefore, the equation of the tangent line is:
.04 yx
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Example 16:
Find the equation of the circle with center (4, 0) and touching
the line Sketch the graph.
.0182 yx
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Analysis:Observe that the only given we have is the center
C (h, k) = C (4, 0) and an equation of a line
If we are to find the equation of the circle we should have the value of radius.
However, we all know that we can find the radius by knowing how far the center C (h, k) = C (4, 0)
to the tangent line .
.0182 yx
.0182 yx
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Something to think about…
If is the formula for finding the distance given
TWO POINTS, then what is the formula for finding the distance given
a POINT and a LINE?
2
12
2
12 )()( yyxxd
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Formula for Finding the Distance Given a Point and a Line:
Formula for finding the distance given a point and a line:
22 BA
CByAxd
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Final Answer:
Therefore, the equation of the circle with center (4, 0) and touching the line
is:0182 yx
.04822 xyx
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Tangent to a Circle:
Tangent in Prescribed Direction
SSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
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What should you Expect?
This section will illustrates how to determine the tangents parallel to the circle given an equation of the
line which is parallel or perpendicular to the tangent line.
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Example 17:
Determine the lines tangent to the circle and
parallel to the lineSketch the graph.
0124622 yxyx
.03043 yx
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Take Note: To determine the equations of the tangent lines
which are also parallel to , we have to use the formula for finding the radius given a point and a line which is . Please take
note that the value of the raduis is ±5 units. Since we are looking for two equations of tangent lines
which are also parallel to ,for special case like this we will use the
negative radius.
03043 yx
03043 yx
22 BA
CByAxr
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Final Answer:
Thus, the equations are
and
02643 yx
.02443 yx
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Example 18:
Given a line and circle find the
equations of the tangents to the circle which are perpendicular to the line. Also, please sketch the graph.
042 yx
,04822 xyx
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Take Note: We need to find two equations of tangent lines to the
circle which are also perpendicular to. Take note that using the Perpendicular
Slope Theorem which states that, “If two lines are
perpendicular, having respective slopes, m1 and m2, the slope of the
line is the negative reciprocal of the slope of the other line,” we can
actually find the slopes of ⊥L2 denoted by m2 and ⊥L3
denoted by m3. If the slope of is
m1 = -1/2, hence m2 = m3 = 2.
042 yx04822 xyx
042 yx
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Take Note: As you can observed, we are to find two equations of
tangent lines to circle which are also perpendicular to using the Slope-Intercept Form since we have m2 = m3 = 2. Thus, we have to find the y-intercepts b1 for ⊥L2 and b2 ⊥L3 . To find this, we have to use the formula for finding the distance from a point
to the line using the formula:
042 yx04822 xyx
bmxy
bmxy
12
11
m
ybmxr
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Final Answer:
Substituting the value of m (slope) and b (y-intercept) in
the equation of the tangents are .
and .
,bmxy 022 yx
0182 yx
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Tangent to a Circle:
Inscribed Circle in a Triangle
SSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
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Inscribed Circle
It is the largest possible circle that can be drawn inside the triangle in
which each of the triangles' sides is a tangent to the circle. We also refer to
this as INCENTER OF ATRIANGLE.
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Incenter
It is the point at which the angle bisectors of a triangle
intersect and it is the center of the circle that can be inscribed in
a triangle.
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Example 19:
A triangle has its sides having equation equal to
and Find the equation of a circle inscribed in a
triangle. Also, please sketch the graph
.092 yx
,02 yx
0162 yx
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Final Answer:
Therefore, the equation of the circle tangent to the three given lines is:
.036104055 22 yxyx
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Tangent to a Circle: Tangents from a Point Outside the Circle
SSMth1: Precalculus
Science and Technology, Engineering
and Mathematics (STEM)
Mr. Migo M. Mendoza
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What should you Expect?
This section illustrates how to determine the tangents that contains a point outside the
circle.
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Example 20:
Find the equations of the tangent line to the circle
from the point (1, 7). Sketch the graph.
046422 yxyx
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Final Answer:
Therefore, the tangent lines are
and
0161247 yx
.01x
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Performance Task 5:
Please download, print
and answer the “Let’s
Practice 5.” Kindly work
independently.