SSMth1: Precalculus - · PDF fileChapter 2: Circles Lecture 2: Introduction to Circles...

Chapter 2: Circle

SSMth1: Precalculus

Science and Technology, Engineering

and Mathematics (STEM)

Mr. Migo M. Mendoza

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

Nice to Know:

TED Ed Video: Why

are Manholes Cover

Round? by Marc

Chamberland

..\..\..\Downloads\Why are manhole covers round - Marc Chamberland.mp4




From TED Ed Video:

Keep your eyes open

and you just might come

across a rule of triangle

manhole.

Nice to Know:

Why is πr2 the

area of a circle?

..\..\..\Downloads\Area of a circle how to get the formula.mp4







Lecture 2: Introduction to Circle

SSMth1: Precalculus



Mr. Migo M. Mendoza

What is a Circle?

Definition of Circle

A circle is a set of all points in a plane whose distances from a

fixed point is a constant.

),( yx

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.

The Derivation of Equation of a Circle

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?

The Distance Formula:

212

2

12 yyxxd

Form of Circle # 1:The Standard Form of an

Equation of a Circle with Radius rand Center at (h, k):

222rkyhx

Something to think about…

What will happen to the standard equation of a

circle with radius r if the center is at the origin?


Equation of a Circle with Radius rand Center at the origin (0, 0):

222 ryx


What will happen to the standard equation of a circle with the center at the origin if the radius is

1?


Equation of a Circle with Radius 1and Center at the origin (0, 0):

122 yx

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

Form of Circle #4:The General Form of the

Equation of a Circle is:

022 FEyDxyx

Lecture 3: Forms of a Circle

SSMth1: Precalculus



Mr. Migo M. Mendoza

Learning Expectation:

This lecture will discuss how to set up the graph of a circle and determine its

radius.

Example 5:

Graph the circle with standard form of:

1622 yx


What do you know about radii of the

same circle?

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.

Did you know?

We can convert standard equation of a circle to its

general form by expanding the binomials using the FOIL

method.

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

Example 7:

Find the general equation of the circle whose center is (2, 6)

and whose radius is 3. Also, please graph the circle.

Final Answer:Thus, the general equation of the

circle whose center is (2, 6) and whose radius is 3:

03112422 yxyx

Performance Task 2:

Please download, print

and answer the “Let’s

Practice 2.” Kindly work

independently.

Lecture 4: Converting General Form to Standard Form of a Circle

SSMth1: Precalculus



Mr. Migo M. Mendoza

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?

A Short Recap:

What is the general form of the equation

of a circle?


What benefit can we get from using standard equation instead of general equation of

a circle?

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.

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

Did you know?

To convert general equation of a circle to its standard

equation, use the completing the square method.

Step 1: Completing the Square

Group the equation according to its

variables.


Ensure that the coefficients of x2 and

y2 are both 1.


Take the half of the coefficient of x and y, squared it, and add it to both sides of

the equation.


Factor by perfect square trinomial.


Simplify the value of the radius (r2).

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

Example 9:Express the given general equation

below to its standard form:

0208422 yxyx


Is there an easiest way to convert the general form of a circle to its

standard form?

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.

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


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?

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.

Example 10:

Express the general form of a circle below to its standard form:

04010622 yxyx


Revisit our previous examples, what have you

observed on the value of the

radius r2? What can you

conclude?

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.

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


Why do you think the value of radius r2 will

never ever be negative?

To sum it up…

What conclusions can we make with respect to the

radius of circle?

Conclusion Number 1:

Whenever the radius of a circle is a positive value,

the circle exists.


Whenever the radius of a circle is exactly equal to

zero, the circle is a point or degenerate circle.


Whenever the radius of a circle is a negative value, the circle does not exist.

Performance Task 3:




independently.

Lecture 5: Circles Determined by Different Conditions

SSMth1: Precalculus



Mr. Migo M. Mendoza

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.

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.


Where can we find the given three

points on the circle?

Method Number 1:

Method Number 1:Determining the GENERAL EQUATION given three points

on the circumference of the circle.

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

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?

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).

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).

Step 4:

Combine the fourth and the fifth equations and solve for the value

of E.

Step 5:

Substitute E = -6 to

the fourth equation to

obtain the value of D.

Step 6:

To obtain the value of

F, we substitute also

E = -6.

Step 7:

Lastly, we substitute

D = -2, E = -6, and

F = 5 to the general equation

of a circle.

Method Number 2:

Method Number 2:Determining the STANDARD EQUATION given three points

on the circumference of the circle.

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

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

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

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

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

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


When can we say a triangle is

circumscribed by a circle?

Tell Me: “Which is Which?”From the two figures, which is

circumscribed circle and inscribed circle?

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.

Performance Task 4:




independently.

Lecture 6: Tangent to a Circle

SSMth1: Precalculus



Mr. Migo M. Mendoza

Did you know?

There are three (3)

saddest love stories in

Mathematics…

The Painful Asymptote

There are people who

may get closer and

closer to one another,

but will never be

together.

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.

The Painful Tangent

Some people are only

meant to meet one

another at one point in

their lives, but are

forever parted.


What are the three (3)

conditions that guarantee

a line is tangent to a

circle?

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.

The Tangent-Line Theorem:

The Tangent-Line Theorem

“If a line is tangent to a

circle, then it is

perpendicular to the radius

at its outer endpoint.”

The Tangent-Line Theorem:


If tangent is a line, what kind of function is a tangent line? Also, what’s its general

equation?

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

Did you know?

There are four main types of problems

concerning tangents to circles.

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

Tangent to a Circle:

Tangent at a Given PointSSMth1: Precalculus



Mr. Migo M. Mendoza

Example 14:

Given the equation of the circle

Prove that is a tangent to the circle. Sketch the graph.

.04514822 yxyx

02 yx


How can we prove thatis a tangent to the

circle ?04514822 yxyx

02 yx

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

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

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

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

Take Note:

“If a line touches the

circle in a single point,

then it’s a tangent.”

Take Note:

“If a line touches the

circle in two points,

then it’s a secant.”

Take Note:

“If a line does not

touch the circle there

is no solution.”

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

Example 15:

Find the equation of the tangent line to the circle

at the point (-2, 1). Sketch the graph.

01710622 yxyx

Five Forms of Linear Equation:

The Slope-Intercept Form

bmxy


The Point-Slope Form

)()( 11 xxmyy


The Two-Point Form

)()( 1

12

121 xx

xx

yyyy


The Intercept Form

0b

y

a

x


The Normal Form

0sincos pyx

The Tangent-Line Theorem

“If a line is tangent to a

circle, then it is

perpendicular to the radius

at its outer endpoint.”

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

Final Answer:

Therefore, the equation of the tangent line is:

.04 yx

Example 16:

Find the equation of the circle with center (4, 0) and touching

the line Sketch the graph.

.0182 yx

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


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

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

Final Answer:

Therefore, the equation of the circle with center (4, 0) and touching the line

is:0182 yx

.04822 xyx


Tangent in Prescribed Direction

SSMth1: Precalculus



Mr. Migo M. Mendoza

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.

Example 17:

Determine the lines tangent to the circle and

parallel to the lineSketch the graph.

0124622 yxyx

.03043 yx

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

Final Answer:

Thus, the equations are

and

02643 yx

.02443 yx

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

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

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

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


Inscribed Circle in a Triangle

SSMth1: Precalculus



Mr. Migo M. Mendoza

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.

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.

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

Final Answer:

Therefore, the equation of the circle tangent to the three given lines is:

.036104055 22 yxyx

Tangent to a Circle: Tangents from a Point Outside the Circle

SSMth1: Precalculus



Mr. Migo M. Mendoza

What should you Expect?

This section illustrates how to determine the tangents that contains a point outside the

circle.

Example 20:

Find the equations of the tangent line to the circle

from the point (1, 7). Sketch the graph.

046422 yxyx

Final Answer:

Therefore, the tangent lines are

and

0161247 yx

.01x

Performance Task 5:




independently.

SSMth1: Precalculus - · PDF fileChapter 2: Circles Lecture 2: Introduction to Circles...

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