Analytic GeometryLecture 2:Circles

Engr. Adriano Mercedes H. Cano Jr.

University of Mindanao

College of Engineering Education

Electronics Engineering

MATH 201

1

Lecture Objectives

Upon completion of this chapter, you should be

able to:

Learn basic concepts about Circle

Plot a circle base on equation given.

Solve problems involving equation of a circle

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Outline

Introduction

Equation of a circle

Graphing a circle

Writing equation of a circle

Forms of equation of a circle

Terminologies

Techniques in solving problems involving

circle

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Circle

A set of points (locus) which are equidistant

from a fixed point called center. The distance

from the center to any points is called radius

Two quantities that are

needed to find the

equation of a circle:

Center

Radius

Equation of a Circle

centered at (0,0)

Example 1

Determine the center and radius of the given equation of circle.

Example 2

Consider the circle below. Fine the

equation of the circle

The center is at the origin To determine the radius:

This leads to the

equation of the circle:

Exercises

Find the equation of the following circles with

center at the origin and:

radius √ 3 units

passing through the point (–5 ; 12)

passing through ( 1/2 ; 1/2 )

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Equation of a Circle

centered at (h,k)

Example 3

Write the standard equation of the circle:

with center at (4, 7) and radius of 5 units

(x – 4)2 + (y – 7)2 = 25

Example 4

Example 5

Determine its center and radius.

by completing the square and factorize

Solution.

Write the standard equation of the circle:

Center (2, -9) Radius of

(x – 2)2 + (y + 9)2 = 11

11

Example 6

Exercise

Determine the co-ordinates of the centre of

the circle and the radius for each of the

following: (x – 3)2 + (y – 2)2 = 9

x2 + y2 – x – 2y – 5 = 0

x2 + y2 + 2x – 6y + 9 = 0

Write the standard equation of the circle with

center at (-3, 8) and a radius of 6.2 units

Exercise

General Equation of a Circle

SUMMARY

x2 + y2 = r2

(x – h)2 + (y – k)2 = r2

Circle

Standard Form

General Equations Form

Terminologies

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Terminologies

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Terminologies

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• A circumscribed circle or

circumcircle passes through all

vertices of a plane figure and

contains the entire figure in its

interior.

• The center of this circle is

called the circumcenter.

Notes!!!

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For a polygon, each side of the polygon must

be tangent to the circle.

All triangles and regular polygons have

circumscribed and inscribed circles.

The radius of the circle is always

perpendicular to the tangent line

Terminologies

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• An inscribed circle is the

largest possible circle that can

be drawn on the inside of a

plane figure.

• For a regular polygon, the

inradius (the radius of the

inscribed circle) is called the

apothem.

• A unique circle inscribed to a

triangle is called the incircle.For a polygon, each side of the polygon

must be tangent to the circle. All triangles

and regular polygons have circumscribed

and inscribed circles.

Problems involving Equation

of a circle

Hints.

Make a rough sketch of the problem

Analyze

Practice

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Points inside, on, outside

a circle

Two Methods

1. Calculate the distance from the center and

compare this distance with the radius

2. Substitute the coordinates into the equations

of the circle.

Example 7

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Excersice

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Intersection of a line and a

circle

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Example 8

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Example 8

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Example 9

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Example 9

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Exercise

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Finding the equation of a

circle: Given 3 points

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Example 10

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Exersice

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Finding the equation of a

circle: Given 2 points and equation of the line

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Example 11

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Example 11

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Cramer’s rule

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Cramer’s rule

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Cramer’s rule

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Cramer’s rule

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Finding the equation of a

circle: Given 2 points and equation of the

tangent at one these points

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Example 12

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Slope of the perpendicular line: - ½

Equation of the perpendicular line:

y – (-2) = (-1/2) (x-(-3))

y+2 = (-1/2) (x+3)

2y +4 = -x-3

x+y+7=0

g+f+7=0 eq1

Example 12

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(-3,-2) on the circle:

(3)^2 +(2)^2 +2g(-3)+2f(-2)+c=0

9+4-6g-4f+c=0

-6g-4f+c=0 eq 2

(0,-1) on the circle:

(0) +(-1) +2g(0)+2f(-1)+c=0

-2f-1+c=0 eq 3

Exercise

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Finding the equation of a

circle: Given a radius, a point and equation of

the line

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Example 13

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Example 13

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Exercise

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Equation of a tangent to a

circle at a given point.

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Example 14

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Proving that a line is a tangent

to a circle

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Exercise 15

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Length of a tangent to a circle

from a point outside the circle

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Exercise 16

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Exercise

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Exercise

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Tangents parallel or

perpendicular to a given line

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Example 17

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Equations from a tangent

outside a circle

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Example 18

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Example 18

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